## Chamok hasan

Chamok Hasan is a Bangladeshi writer and mathematician. He and Hasibul Ahsan were in same department in buet. They are known to each other for long time.

Chamok Hasan has a good skill of teaching mathematics.

## Chamok Hasan calculus

Limit, after all, is such a concept which was developed using logic. The defintion of limit was rooted in the idea of converging quantities:
If a quantity approach another quantity making the difference between them less than any arbitrary small number in some interval of time then they become ultimately equal at the end.
This was Newton's idea of converging numbers. But he was never able to base calculus on perfect definition of limit. He knew calculus worked but never questioned about its validity. Later a mathematician named Cauchy defined limit mathematically. Limit is the value which some function approaches but never reaches it. For more explanation follow this page.
Mathematics follow logic. It took a long time to establish connection between mathematics and logic. The most notable endeavour to find such connection was undertaken by Bertrand Russell and Dr. Alfred North Whitehead. Unlike we arevery mistaken, all mathematics are logical deduction from a very few logical premises. Pure mathematics has shown that ten logical premises and almost twenty principles of deductions are enough to define whole of mathematics from algebra to geometry. This is how it goes.
Pure mathematics contains no constant except logical contant and consequently no premises, or indemonstrable propositions, but such as are concerned exclusively with logical constants and variables . It is precisely this that distinguishes pure mathematics from applied mathematics. In applied mathematics, results which have been shown by pure mathematics to follow from some constant satisfying the hypothesis in question. Thus terms which were variables become constants, and a new presmise is always required, namely: this particular entity satisfies the hypothesis in question. Thus for example Euclidean Geometry , as a branch of pure mathematics, consists wholly of propositions having hypothesis "S is a Euclidean space" . If we go on to : "The space that exists is Euclidean" this enables us to assert of the space that exists the consequents of all the hypothetical constituting Euclidean Geometry, where now the variables S is replaced by the constant "actual space". But by this step we pass from pure to applied mathematics.
Implication of pure mathematics can be put in a truth table like this :

The gemoetrical interpretation of calculus can be put like this:

The derivative of a function at a point x is the slope of the tangent drawn on that point. So tangent is actually defined through differentiation. It always finds the rate of change of some function f(x) at specific value of independent variable x. For more explanation vist this page.

## Integration

Integration is the inverse process of differentiation. It is a process of summing infinite number of quantities to find the area under a curve. We can define integration in the following way:

If we diffentiate the integral then we get the original function back. This is called the fundamental theorem of calculus.
Some more integration formulas are given :

We sometimes integrate an argument by substitution.

## Integral equation

Integral equation is quite similar to differential equation. Integral equation contains an integral of some function. A general form of Fredholm Integral Equation is

An integral equation is an equation in which an unknown function appears under one or more integration signs. In the above equation K (x,t) is called the kernel. Fourier transform is an example of integral equation where the Kernel is the exponential function e^(-2ix). Fourier transform is sometimes called the integral transform.

## Functional derivative

Functional is a function of a function . We can define a rule for differentiating functional like the ordinary differentiation of function.

A more detailed explanation involves variational method

A formal representation would be like as follows:

An example of a functional derivative can be given here.

We treat the function as if it were an independent variable.

## Arithmetic

Arithmetic is a mathematical system which is concerned with addition , subtraction , multiplication and division of numbers. Artithmetic must satisfy these properties concerning numbers

## Fuzzy Logic

In usual logic like boolean logic the truth value of logical statemenet can only be either zero or one. It is equivalent to switching on or switching off. In fuzzy logic the truth value can take any value ranging from 0 to 1. That is to say it can be .4 or .5. Google search algorith uses fuzzy logic everytime to return the most relevant results of the searched keywords. The set of truth values can be anything in between 1 and 0 .

## Fibonacci Sequence and Golden ration

Fibonacci sequence is the sequence where any number in it is the sum of previous two consecutive numbers just before it.

Two quantities are in a golden ratio if it is the same as the ratio of their sum to the larger of the two quantities.

## Munchausen Number

It is one of some weird numbers , which has special property:

## Series and sequence

There are two kinds of series and sequences : one is arithmetic series and the other is geometric series. In arithmetic series there is a constant difference between two consecutive terms and in geometric series there is a constant ratio between two consecutive terms. There are special rules for summing series. These are as follows :

There is another type of series named harmonic series. If you take reciprocal of every term of an arithmetic progression you get a harmonic series. It is a divergent infinite series. Every term after the fast term is the harmonic mean of the neighboring terms.
The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations.
An example of harmonic series is :

There is a simple proof that harmonic series diverges. The proof goes like this :

Musical interval and notes are related to the terms of harmonic series:

The first note leveled at one(1) is the double of second note 1/2, which is called an octave. An octave increases the frequency of a note to the double of that. The difference of the note 3 and 4 is called perfect fourth. It is a difference of frequency (1/3 - 1/4) = 1/12 of the frequency of first note ( it may be 350 hz). Similary various intervals comprise of various musical intervals. All have specific names. Music is nothing but precision of harmony.

## Continued fraction

Continued fraction is a special type of fraction where the denominator repeats over itself. It looks something like this:

## Logarithm

Logarithm was invented by Napier. We all know about Napier's scale. It is a scale which turns multiplication into summation. How is that possible? well it is based on rules of logarithm. Lets first define logarithm. If a to the power x is equal to y than we define log (a) y = x . Usually we use 10 base logarithm. That is to say if 10 to power x = y then log(10) y = x .
If we muliply to logarithm log(y) and log (z) we get log(y) X log (z) = a + b where 10^z = b and 10^y = a . New scale will read a+b . This is easy to verify.
Now suppose there is two scale which reads logarithmic value of 10^a and 10^b respectively corresponding to values of a and b. Now if we multiply the two scales we get a+b as a new scale. Thus logarithm can help us convert multiplication into addition. This was John Napier's invention which is still very useful. What is scale? scale is the ratio of one quanity to another. The time scale according to relativity is different at different location in space around the whole universe. This is a topic to be discussed in Theory of relativity.
Logarithm of some variable can be expanded as power series. Here are some examples:

## Algebra

Algebra is perhaps the most important tool in mathematics. We can not solve any equation without the rules of algebra. But what is algebra?
Algebra deals with variables which obeys the rules of arithmetic. There is a basic difference between algebra and arithmetic. Algebra is exclusively concerned with variable which can take many numberic value while arithmetic is concerned with constants and numbers. And of course there are rules of subtraction, multiplication, addition and division which are defined in a certain way.
A very basic equation of algebra can be mentioned :

So an equation involves one or more variables and for some value or values of the variable satisfy the equation. Other meaning of equation can be that it expresses an equality. Now let us go back to set theory.
Solving exponential equation is somewhat different. Exponential equation contains exponent of some variables. This is the way exponential equation is solved:

You can watch this video of Chamok Hasan on algebra for elementary ideas:

My video on pure mathematics

Algebra , by dint of equations, helps us solve many real world problems. We can turn many quantitative problems into algebra and solve the equations for understanding. For example, if you are given that the interest rate of your bank account is compounded interest and you deposited certain amount of money. How much money will you get after certain amount of time? Compound interest means that your amount of interest will change over time based on your current holding, which includes all the previous accumulated interest based on your deposit.

This is the kind of problem algebra deals with exclusively. There are many harder problems. All problems are just converting the variables into numbers and solve for unknown. The first successful marriage between algebra and geometry was done by Rene Descartes. Descartes named it coordinate geometry.
Now we come to the notion of constant. Variable is some term in mathematical statement, that does not change it value. But what does it mean to not change its value? we can consider a class of straight lines given by the equation y = ax + b . As long as a and b are fixed we only get only one staight line. But for different values of a and b we have different straight lines on a plane. Are a and b necessary a constant? It is as long as we consider one straight line but they become varaibles when the above equation represents a class of straight lines. Similarly value of π is constant as long as we consider all the circles in Euclidean space. It will be different in different curved spaces.

## Vector algebra

Vector algebra is a very useful branch of mathematics which deals with the vectors and its properties. It is useful in physics also. Vector is a quantity which has both a direction and a magnitude. Some elementary vector algebra equations are given below :

Cross product of two vectors are defined in the following way:

It is a vector which also has a magnitude and a direction.
Linear algebra is the branch of mathematics , which deals with vectors and matrices. It is very useful in both mathematics and theoretical physics.
A set of vectors is said to be linearly independent if the following conditions hold.

## Eigen Value

Eigenvalue is the which , when muyltiplied by some quantity equals some operator acting on the quantity. For example , some operator T acting on a vector v produces a scaled version of v. Mathematically it is
T (v) = λv
In this case , v is called eigenvector. Similar condition can be apllied to functions as well.

Where I is the identity matrix.

A block matrix is a matrix which has other matrices as its elements.

## Singular value decomposition

Singular value decomposition is a factorization of real or complex matrix. Suppose M is an m × n matrix whose entries come from the field K, which is either the field of real numbers or the field of complex numbers. Then the singular value decomposition of M exists, and is a factorization of the form
M = U∑V* Where V is the complex conjugate transpose of V.
where U is an m × m unitary matrix over K , unitary matrices are orthogonal matrices),
Σ is a diagonal m × n matrix with non-negative real numbers on the diagonal,
V is an n × n unitary matrix over K, and V∗ is the conjugate transpose of V.
The diagonal entries σi of Σ are known as the singular values of M. A common convention is to list the singular values in descending order. In this case, the diagonal matrix, Σ, is uniquely determined by M (though not the matrices U and V if M is not square).

A graphical illustration of singular decomposition of some mXn matrix is given. Multiplaction of three different actions U, ∑, and V* can immitate the complete action of singular value decomposition.

## Matrix function

Matrix function is defined using Taylor series. Here the independent variable is matrix. It is a function which takes a matrix and outputs another matrix.
Tylor polynomial is can be written as :

If we replace matrix A for x then addition becomes matrix sums and multiplication becomes scaling operations. This way we can find exponential of any function.
If the input matrix is diagonalizable : That is to say, A = PD(1/P) here 1/P is actually inverse matrix of P, then f(A) is defined by

## Cayley Hamiltonian theorem

Cayley Hamiltonian theorem is a proposition of matrix algebra which says the a square matrix satisfies its own characterstics equation. The statement can be stated as below:

Statements of algebra have precise interpretations as they convey truth about the consequences of some hypotheses of pure mathematics. For example we all know 1+1 = 2 bears a true fact. But what does this mean? It is a mathematical statement which is always true. Bertrand Russell and Whitehead proved this statement using a long argument.

Let us interpret this statement in context of pure mathematics. In context of pure mathematics it states that " if both x is one and y is one and x is not y then x and y are two". It also means " if a is not b then whatever x may be , x is an a is always equivalent to x is b or x is γ ". Now we have no numbers but only variables and propositions.
Some basic algebra equations are related to bionomial theorem :

As I mentioned bionomial theorem it is better to give its formula explicitly

Equations or formulas form the backbone of algebra and consequently of all the mathematics. Without formula we could not solve anything and compare one thing with other things. Bionomial theorem needs the concept of permutation and combination.
Various components or parts of an equation can be identified like this :

Permutation is the different arrangement of specific number of things, that we can make out of some given number of things. Suppose we have three letters A, B and C. If we are to ask how many ways we can arrange all those letters or some of those then we need the formula of permutation as follows:

In permutating three letters A, B and C we come up with six different arrangements. That is the result obtained from 3! = 6 but if we want to make a combination of A, B and C we get only one such collection of A, B and C. So to get the result of combination we divide the result by the of permutation by the factorial of the number of things we want to combine. The order of permutation in the combination does not matter. That is the basic difference between permutation and combination and that is the reason why r! is included in the denominator of the combination formula.

## Laws of thought

Goerge Boole first developed a system for logical treatment of any number of propositions having truth values. These truth values are normally zero or one. Modern computers have been developed using this system that George Boole invented. It is best known as boolean algebra. Boole named it laws of thought. Laws of thought obeys these three rules of logic :

## Linear Algebra

Linear algebra is the branch of mathematics , which deals with the linear equations. We can use matrices to solve a system of linear equations. This is called Cramer's rule :

## Diophantine equation

In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied. A linear Diophantine equation equates the sum of two or more monomials, each of degree 1 in one of the variables, to a constant.

## PEDMAS

The rule of simplifying algebra problems is a special method named PEDMAS. It means P = parenthesis, E = exponential, D = division, M = multiplication, A = addition, S = summation. That is say, we always maintain this order in simplifying algebraic expression. Take a look at this example:

## Number theory

Mathematics is the queen of science whereas number theory is the queen of mathematics. Number theory is concerned with properties and relationship of numbers , expecially the positive integers. Euclid first developed an algorithm to compute greatest common divisor of two integers. This was the first application of modern number theory. The algorithm can be stated as follows:
suppose we want to calculate GCD of two integer 64, 12 . We fist check the remainder of 64 and 12 when dividing the fist number by the second number.
64/12 => 4 as 5X12 = 60
Now divide 12 by 4 to check whether there is any remainder. We get a remainder of 0 when doing that. So the greatest common divisor of 64 and 12 is 4. When we get 0 as a remainder we finish our division. We can check that by finding the gcd of 64 and 12 in usual way.
Factors of 64 are 4X8X2 = 2X2X2X2X2X2 and factors of 12 are 2X2X3 . So the factors that are in common in 64 and 12 are 2, 2 which means 2X2 =4 is the greatest common divisor of 64 and 12.

## Chineese Remainder theorem

The Chinese remainder theorem is a theorem of number theory, which states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime.
If you are given certain number of things and you are told them that when you count them by seven six things left over, if you count them by five four things left over and if you count them by three two things are left over. What is the actual number things? This kind of problem needs to be solved using the chineese remainder theorem.
Statement algorithm of chinese remainder theorem:

# Quantum Field Theory

Feynman's sum over histories   |   S-matrix   !   quantum field theory

# Theory of relativity

Relativity made simple   |   Special theory of relativity   |   General theory of relativity   |   Tensor calculus  |   Hamiltonian mechanics   |   Field equation  |   Perihelion of mercury|   Geodesic distance

# Quantum mechanics

Schrodinger equation   |   Matrix mechanics   |  Dirac equation

## Linear equation

A linear equation is the equation where the dependent variable changes with the independent variable linearly. That is to say, if we plot the graph of such equation, we get a straight line. General forms of linear equation are as follows:

## Absolute Value

Absolute value of a number x is defined as |x| = x if x is positive and |x| = -x if x is negative . If x is a real number then |x| <= x

## Complex number

Complex number is written in the form z = a + ib where a, b are real numbers and i = square root(-1 ). There are some identities and inequalitites related to modulus of complex number. These are as follows :

Where modulus of the complex number is defined as :

Complex analysis is perhaps one of the most intriguing topics in mathematics. It has a large class of applications in subjects like electricity , quantum electrodynamics and quantum field theory.

## Euler Identity

Euler Identity can be proved in the following way:

## Set theory

Famous mathematician Goerge Cantor first gave the concept of set. Set is any collection of individuals. It is a class in some sense. For example class of people of Bangladesh can be regarded as set. Class of addresses in Dhaka city can be considered a set. It is not necessary that set must contain real objects. Set can contain abstract elements. I remember now Russell's paradox. Russell found that some class contains itself. For example class of all abstact concepts is also a concept. But most of the classes are not members of themelves. Like the class of all bicycles is not bicycle itself. This also applies to sets. Then Russell formed a class of all the classes which does not contains themselves. He thought also if this class contains itself or not. If this class contains itself then by definition it follows that it does not contain itself. Again if it does not contain itself then it contains itself. So we come to a contradictory situation. Russell spend a long time to resolve the paradox and at last he proposed the theory of type which he later abandoned as a failure.
Set is too useful concept in mathematics. It has turned out to be eventually. We can expresses numbers in terms of sets , particularly empty sets. Empty set is a set which has no element. Now consider an set which has an empty set as its element. Like this : φ = {} , A = {φ , φ }
Continue this up to sets with three and four elements and five elements..
B = {φ, φ, φ) , C = {φ, φ, φ, φ} and D = {φ, φ, φ, φ , φ} and so on .. We can thus express numbers without pointing anything that is physical. This is a simple example of how much powerful the concept of set can be.
Another example of empty set can be found in the definition of ordinal numbers. Ordinal numbers can be defined as class of all well-ordered sets. For example ordinal 6 will be class of all sets which contains 1,2,3,4,5,6 . But instead of defining ordinal as class of well ordered sets , it can be defined as a particular well-ordered-set which cannonically represents the class. For example, we can set:
0 = {} = φ
1 = {0} = { φ}
2 = {0,1} = {φ, {φ} }
3 = {0,1,2} = {φ, {φ}, {φ {φ}} }
and so on. Each set will represent a specific ordinal number. Ordinal arithematic is much harder and at the same time interesting. Cardinal arithematic is easier. These two kind of numbers namely ordinal and cardinal were defined by Cantor for the first time.

## Geometry

Geometry is perhaps the most important branch of mathematics. It is also the highest exercise of human brain. We all know Euclid's book named "elements". In elements Euclid has proved lots of propositions and theorems of his geometrical system known as Euclidean geometry. But what is this geometry actually. Euclid defined some terms and gave few axioms based on which he developed his geometrical system. His axioms were self-evident truth. One of his axioms is like "from one point to another only one straight line can be drawn". Bertrand Russell was once being taught a course on Euclidean geometry by his tutor. His tutor told him to accept the axioms as true without any objection. But Russell refused to accept those at first. His tutor retorted if you cannot accept those we cannot go on. So Russell had no other choice than to accept those as true and go on. In logic we do not accept such method. When mathematics is blended with logic , no such self -evidence truth is aceptable. What we can only assert is implication. P implies Q means either P is true or Q is false. P and Q are here propositions. So in mathematics we do not actually know what we are talking about nor that it is actually true. It only assures us that if such and such thing are true of something then such and such other thing are also true of the same thing. Yet mathematics can be very powerful scientific branch to search for the secrets of the nature. By the way we now get back to Euclid's elements.
Euclid's "elements" has thirteen books. Book 1 contain five postulates and 5 notions. Book 2 contains number of lemmas (a subsidiary or intermediate theorem in an argument or proof) concerning rectangles and squares. Euclid proved a whole bunch of propositions and theorems which are now parts of elementary geometry books.
What is the definition of parallel line according to Euclid's ideas?
Given a straight line , there are two classes of lines. All the lines corresponding to one class cut the straight line and lines belonging to the other do not. There is a limiting class between these two classes. This is called the class of parallel lines with respect to that given line. In Euclid's geometry given a point and a line lying not on the point there is only one straight line parallel to it. And parallel lines never meet with each other even at infinity. But in non-Euclidean system there are many lines through a given point , which are all parallel to a given line. Non-Euclidean geometry is counter-intuitive although it only asserts implications like in Euclidean geometry. We can not visualize curved space but it exists. Einstein showed us that.
There are other geometries like affine geometry, metrical geometry, descriptive geometry , projective geometry and others. In metrical geometry only spaces equipped with a metric is dealt with. Metric allows us to make distance measurement. What do we mean by distance in mathematics? Distance is a relation between points. In two dimensions distance is a function of four coordinates (x1, x2, x3, x4) which we can deduce by applying Pythagoras Law. Distance only asserts quantitative property. It is not necessary that distance is what we mean by 5m(five metres). It can be 5s (give seconds) too.

## Affine space

Affine space is a geometric structure that generalizes some of the properties of Euclidean space in such a way that is independent of the distance and angle measurement, keeping only the properties related to parallelism and ration of lengths of parallel line segments. In affine space there is no distinguished point that serves as the origin. It a vector space with origin removed. This let us give the concept of affine function. It is a function the form f(x) = ax + b where a and b are constants.
This function may serve as the transformation rules for affine space where set of points are distinguished by this special rule. Geometrically affine transformation preserve the ratios of distances along parallel lines. Affine geometry is the geometry of this affine space. Any geometry in general is a set equipped with a set of transformation laws. In euclidean geometry the transformation rules are such as to preserve the distance between points.

In affine geometry one uses the Playfair's axiom to find parallel lines through some point. And Playfair's axiom states:
Given a point and a straight line one can one line parallel to that given line. This axiom is very similar to Euclid's fifth axiom. In other words Playfair's axiom implies Euclid's fifth axiom.

## Polynomial equation

A polynomial equation is an equation containing linear combination of various powers of an unknown variable x as follows:

Polynomian equation relates fundamental theorem of algebra. It says that a polynomial equation of nth degree having a complex coefficients has at least one solutions. This includes polynomials with real coefficients too since real coffiecients are the complex number having imaginary part equal to zero.

## Probability

Probability is the ratio of two numbers. It is the ratio of the number of way a specific event can happen and the total number of events that can happen. The exact expression would be :

## Conditional Probability

So the general mathematical formula would turn out to be :

There is some misconception about conditional probability. Let me phrase one :
Someone hears that the chance that a single bomb placed in a plane is 1/10. He was a psychic patient and thought that the bomb has a chance to explode. Now he carries another bomb while he travels by the plane. He thought that the chance of explosion would be less. But actually he was wrong.
The fallacy was not to see that the chance that both the bomb being exploded in 1/100. In fact the chance that a single bomb being exploded is still the same and it was 1/10.
Byesian model
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes’ rule) describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if cancer is related to age, then, using Bayes’ theorem, a person's age can be used to more accurately assess the probability that they have cancer, compared to the assessment of the probability of cancer made without knowledge of the person’s age.
Mathematically it can be forulated with the following expression:

This is almost similar to conditional probability theorem but can be used in more complicated cases.

## "Know the formula and solve the problem"

Any quadratic equation can be solved using a closed form formula. This is a well known formula which any school student know more or less. This is as follows:

## Projective geometry

Projective geometry comes from , what the name suggests , projection. It deals with all the properties of geometrical figures that are unaltered by continuous projections. It is not concerned with the quantitative aspects but rather qualitative aspects of lines and points. Projective coordinates are assigned in projective space.

In the figure above, an example of projection from a point is illustrated. Point A, B, C, D are projected into points A`, B`, C` and D`. In this projection certain quantity named anharmonic ratio remains constant. It is the ratio of different line segments formed by the points A, B, C and D.

There is a concept in projective geometry , called duality. The principle of duality says that two distinct points determine a unique line (i.e. the line through them) and two distinct lines determine a unique point (i.e. their point of intersection). This is a proposition of projective geometry. This is how it is so :

A ray is one part of a straight line. So from point A a ray can be drawn extending it to infinity. In the figure alluded to above , from point of origin O all the rays are shown extending from it. The ray may be thus taken as an asymmetricalrelation, or as the half-line on one side of a given point on a line. The concept of ray is very important. It can be used to define angle. What is an angle?
Elementary geometry assumes that two rays starting from the same point determine a certain magnitude, called the angle between them. This magnitude can be defined in several ways. Since the rays in a plane through any point form a closed series, every pair of rays divide the plane into two strethes of rays. These stretches can be defined as the angle between them. As stretch, in general , is number of terms between any two terms of a series.
Sometimes concepts of pole and polar are important in geometry , especially in studying conic sections. The pole and polar are respectively a point and a line that have a reciprocal relationship with respect to a given conic section. The reciprocation of a circle means every point in the plane into its polar line and each line in the plane into its pole. Here is an example

The polar line q to a point Q with respect to a circle of radius r centered on the point O. The point P is the inversion point of Q; the polar is the line through P that is perpendicular to the line containing O, P and Q

## Twistor space

In Twistor theory twistor space is a projective space of spacetime coordinates. A space-time point can be represented as a Riemann sphere in terms of some section of its light cone. This is precisely how space-time points are represented in the projective twistor space.

A full twistor space is a four-dimensional complex vector space.

The twistor space is called projective space because of the duality that arises between Minkowski space and Twistor space. Duality in projective geometry states that lines are equaivalent to points and points are equivalent to lines.

## Partial derivatives

Partial derivative is needed to find the differentiation of multi-variable function with respect to one of the variables of which the function is differentiable. In partial differential equation there is relation of a function to its partial derivative. Partial derivative is defined in the following way:

All the higher mathematics is basically calculus and mathematical analysis. The later deals exclusively with limits, differentiation and integration of analytic function. If you can master calculus you can master a great deal of advanced mathematics.

## Trigonometry

Trigonometry is a branch of mathematics which deals with triangles and its angles. First it is better to define angle. What is angle? Angle is the relation between two straight lines. That is to say, when two straight lines meet they create an angle between them. The common point they share is called the vertex. It is better to replace straight lines with rays. Rays are any line extending from a point to infinity. The plane where the angle is created does not need not to be Euclidean plane. When two rays are perpendicular they create a right angle between them.
Solid angle is defined using a sphere in 3D space. A total of 4π steredian is subtended from the center of the sphere. A sphere has a total of 4πr^2 area. So there is 1/r(square) steredian per unit area of a sphere.
These are the basics of trigonometry. In addition we divide the plane with four quadrants at the origin(0,0). The main ingredient of trigonometry is the triangle.

Cosine is the ratio of adjacent to hypotenuse of a tright triangle. The side which makes the angle with the hypotenuse is the angle in consideration. The opposite side of the right angle is called hypotenuse.
Sine is the ratio of hypotenuse to the opposite or perpendicular.

## Asymptote

Asymtote of a curve is the line such that the distance between the curve and the line approaches zero as one or more of the coordinates x, y tend to infinity.

As an example an equation of asymtote can be given :

We can see that the distance between the line y= x+1 and the curve become zero as the corrdinate x tends to zero in both side.

## Functions

Function is the most important concept in Mathematics. A function is a relation or mapping between two sets. In mathematics , we are always concerned with numeric functions. But in principle, functions other than numeric are also possible. A function is defined from one set which is called domain to another set which is called co-domain. So a function maps elements from doamin to co-domain. Range is the value b = f(a) which is an element of co-domain. Not every element of co-domain is the element of range.
Suppose a function f : inhabitants -> R+ finds the age of inhabitants of a country. The age is a positive real number. If there are 30 million people in america we can find the age of all those inhabitants according to this function. Here all the ages of the inhabitants of america form the image of the function. Image is sometimes called the range of the function too. One-one function is a function which map exactly one element of domain to exactly one element of co-domain. That is , if x1, and x2 is two elements of domain then f(x1) != f(x2). One-one function is also called injective function too.
f(x) is called surjective if the co-domain equals the image. That is to say, f(x) equals Y (co-domain). In other words, pre-image 1/f(y) is not empty. Every element of the co-domain is mapped to some element of the domain.
A function is called bijective of it is both injective and surjective. Function f(x) is bijective if the pre-image 1/f(y) contains exactly one element. It is also called a one-to-one correspondence.

## Parametric equation

Parameter is an independent variable that can be used to represent other variable. Suppose we have an equation of a circle involving two variables x and y. This equation involves only two variables. The equation can be explicitly written as
x^2 + y^2 = r^2 where r is a constant.
Now we can represent the same equation in terms of a third variable named θ Writing x = rcosθ and y = rsinθ we get the same equation of a circle. Another example of parametric equation is

Parametric equations are very important in pure mathematics and mathematical physics. Sometimes manifold is defined using parametric equation , where every coordinate is a function of some parameter.

## Generating function

Generating function is a way of encoding infinite sequence of numbers( a[n]) by treating them as coefficients of a power series. This formal power series is the generating function.
A generating function is like a bag. Instead of carrying many objects detachedly we put them into one bag. Then we would have only one object to carry. Some example of generating functions are :

## Math formula cheat sheet

All the major equations can be put into a package for better correlation between various parts.

Some trigonometric identities are important.

## Statistics

Statistic is the method of collecting a vast number of data and do various kinds of calculations on them. Most useful terms of statistics are median, mode and mean. Here are some useful equations of statistics.

## Histogram

Histogram is an accurate distribution of data in numerical form. It provides us with the probability distribution of a continuous variable. Suppose you have some data about the height of a population. Histogram will tell you how much height is possesed by how many people. To construct a histogram, the first step is to "bin" (or "bucket") the range of values—that is, divide the entire range of values into a series of intervals—and then count how many values fall into each interval.

## Body mass index

Body mass index is a quantity to measure the effective weight of a person, that is considered to be appropriate for that person. It has a mathematical formula given by the equation

## Prime polynomial

Some polynomials can not be factored. This kind of polynomials are called prime polynomials. An example is given below:

## Linear Transformation

Linear transformation T(A) = B where A and B are vectors. T is some opertor or matrix if these two conditions hold:
1. T(a + b) = T(a) + T(b).
2. T(c.A) = cT(A)
Here is an example of a linear transformation :

## Big O notation

We always encounter this big O notation in mathematics. This notation was introduced by the brilliant mathematician Landau, and has been found to be very useful in analysis and in number theory. Let g(n) be a given function of n defined for all n=> a where a is some positive integer. If there exist a simple monotonic positive function h(n) defined for all n =>a and a constant k independent of nn such that
|g(n)| <= kh(n) for all n => a then we write
g(n) = O(h(n)).
Note that k may be any real positive number but in what follows we shall always assume k to be a positive integer. h(n) is usually a simple function such as
n, logn, nlogn, 1 , ..

## Dimensional analysis

Dimensional analysis is the study of the relationship between various physical quantities by identifying their base quantities ( mass, length, time) and units of measures and tracking their dimensions in calculation and comparisons. So for example knowing the dimensions of the base quantities like mass, length and time we can find the dimension of force , work and other quantities. When we have an equation involving multipe terms their dimensions must be identical.

## Subspaces

Sometimes concept of subspace of a vector space is necessary. A subspace can be defined in the following way:

## Gardient descend

Gradient descend is an algorith in machine learning. Machine learning is a branch of computer science and related to artificial intelligence(A.I). Gradient descend involves a cost function to be minimized. So it also involves calculus of variation. To do this we need to start with a cost function

Where m is the number of training examples, θ is the chosen parameter of weights (θ0, θ1, θ2) and x(i) and y(i) are the input and output levels for ith training respectively. We want to minimize this function. Thus the name "gradient descend" appears. The minimum value of this function makes it way down to the lowest point of the graph. The cost function is a function of variable θ . By differentiating the function independently with respect to θ0 we get

Similarly we can do it for the other weight &theta1; too to find the required sesults.

## Application of calculus

Here is application of calculus to find the volume of a cylindrical region enclosed by a three dimensional tactrix.

some imortant formulas of limit are given as well

## Some real analysis

Real numbers are the heart of mathematics. Without real number continuity would not be defined , hence no calculus would have been developed. So some basic understanding of real numbers should always be made. I have given a short overview of real numbers here. Now some basics properties and lemmas of real numbers will be discussed. First supremum of a set is the greatest lower bound. Supremum can be used to define integration too.

## Spring and mass system

A spring motion can be modeled with differential equation. The restoring force is proportional to x and negative. As a result the equation of motion can be written using Newton's second law of motion

Where k is the spring constant. Some important trigonometric formulas are

There is a lot of use of logarithm in music

Some identities and formulas of geometric series must be remembered

Example of geometric series is

We can apply functions to find the terms in a sequence

The explicit formula for the sequence can also vbe given

## Descartes solution of quadratic equation

Descartes found a geometric solution of quadratic equation

Now we return to elliptic curve. All elliptic curves are given by the equation

Such elliptic curves has only finite numbers of solution in integers but can have infinite numbers of rational solutions.

## The logistic model of population growth

Estimates the change in a population of creatures across generations with limited resources. The equation of logistic model is

Analytic countinuation of the factorial of n (n!) can be done. The usual law of factorial n! = n(n-1)(n-2).. works only for integers. After analytic continuation is made the fomula works for positive and negative integers, fractions and even for complex numbers. The analytic continued function of n! is
The same integral for z+1 is defined as the gamma function which has a lots of applications.
Fibonacci sequence can be generated using a function involving golden ration φ. Fibonacci sequence is a sequence of numbers where a number in the sequence is the sum of other two consecuitive numbers prior to it. This is the function

## Chaos Theory

“A butterfly flaps its wings, and it starts to rain,” the narrator of one episode of How I Met Your Mother begins begins, “It’s a scary thought but it’s also kind of wonderful.” He continues:
“All these little parts of the machine constantly working, making sure that you end up exactly where you’re supposed to be, exactly when you’re supposed to be there. The right place at the right time.”
It’s depicted differently in each one, but you see a popularized version of the Chaos Theory everywhere in movies and TV shows – a butterfly flaps its wing, and the course of history is changed forever. The theory isn’t as crazy as you would think.
Traditionally, scientists believed that all natural processes were either deterministic or nondeterministic – implyingg we can either predict their behavior, or not at all. Throwing a ball is deterministic, because if you throw a ball at exactly the same angle and speed, you can predict how far it goes. Uranium decay is nondeterministic, because it’s impossible to predict which exact atom will shrivel at a given time.
Mathematicians later discovered that those two categories aren’t enough (are they ever satisfied?) Some processes seem predictable, but small changes would bring catastrophic consequences. For example, consider a double pendulum swinging wildly. The locations of the two joints seem easy to calculate, but can vary wildly with the initial acceleration and position.

Another example of chaos is the Lorentz attractor. It is a solution of three body problem. The solution tends to converge to two sets as depicted below:

This is better known as the butterfly effect. There is a famous movie named "butterfly effect" which may or may not be related to this concept of chaos. But the movie is awesome and thrilling. The chaotic system is very sensitive to initial conditions.

## Cantor's Theorem

2^|S| > |s| where |S| is the cardinality of the set S. If S is a finite set then its cardinality is the number of elements in it, and things are not very interesting. But the concept of cardinality makes sense also for infinite sets. The power set of a set is the set of its subsets. It is easy to see that for finite sets S the cardinality of the power set equals 2^|S|. Thus we denote by 2^|S| the cardinality of the power set even for infinite sets S. Cantor's Theorem states that the cardinality of the power set of a set S always exceeds the cardinality of S itself. That's obvious for finite sets but far from trivial for infinite sets.

## Boltzman Entropy

A key equation for statistical mechanics formulated by Ludwig Boltzmann. It relates the entropy of a macrostate (S) to the number of microstates corresponding to that macrostate (W). A microstate describes a system by identifying the properties of each particle, this involves microscopic properties such as particle momentum and particle position. A macrostate designates collective properties of a group of particles, such as temperature, volume and pressure. The key thing here is that multiple different microstates can correspond to the same macrostate. Therefore, a simpler statement would be that the entropy is related to the arrangement of particles within the system (or the 'probability of the macrostate'). This equation can then be used to derive thermodynamic equations such as the ideal gas law.

## Cantor's Theorem

2^|S| > |s| where |S| is the cardinality of the set S. If S is a finite set then its cardinality is the number of elements in it, and things are not very interesting. But the concept of cardinality makes sense also for infinite sets. The power set of a set is the set of its subsets. It is easy to see that for finite sets S the cardinality of the power set equals 2^|S|. Thus we denote by 2^|S| the cardinality of the power set even for infinite sets S. Cantor's Theorem states that the cardinality of the power set of a set S always exceeds the cardinality of S itself. That's obvious for finite sets but far from trivial for infinite sets.

## Boltzman Entropy

A key equation for statistical mechanics formulated by Ludwig Boltzmann. It relates the entropy of a macrostate (S) to the number of microstates corresponding to that macrostate (W). A microstate describes a system by identifying the properties of each particle, this involves microscopic properties such as particle momentum and particle position. A macrostate designates collective properties of a group of particles, such as temperature, volume and pressure. The key thing here is that multiple different microstates can correspond to the same macrostate. Therefore, a simpler statement would be that the entropy is related to the arrangement of particles within the system (or the 'probability of the macrostate'). This equation can then be used to derive thermodynamic equations such as the ideal gas law.

## Cantor's Theorem

2^|S| > |s| where |S| is the cardinality of the set S. If S is a finite set then its cardinality is the number of elements in it, and things are not very interesting. But the concept of cardinality makes sense also for infinite sets. The power set of a set is the set of its subsets. It is easy to see that for finite sets S the cardinality of the power set equals 2^|S|. Thus we denote by 2^|S| the cardinality of the power set even for infinite sets S. Cantor's Theorem states that the cardinality of the power set of a set S always exceeds the cardinality of S itself. That's obvious for finite sets but far from trivial for infinite sets.

## Boltzman Entropy

A key equation for statistical mechanics formulated by Ludwig Boltzmann. It relates the entropy of a macrostate (S) to the number of microstates corresponding to that macrostate (W). A microstate describes a system by identifying the properties of each particle, this involves microscopic properties such as particle momentum and particle position. A macrostate designates collective properties of a group of particles, such as temperature, volume and pressure. The key thing here is that multiple different microstates can correspond to the same macrostate. Therefore, a simpler statement would be that the entropy is related to the arrangement of particles within the system (or the 'probability of the macrostate'). This equation can then be used to derive thermodynamic equations such as the ideal gas law.

## Cantor's Theorem

2^|S| > |s| where |S| is the cardinality of the set S. If S is a finite set then its cardinality is the number of elements in it, and things are not very interesting. But the concept of cardinality makes sense also for infinite sets. The power set of a set is the set of its subsets. It is easy to see that for finite sets S the cardinality of the power set equals 2^|S|. Thus we denote by 2^|S| the cardinality of the power set even for infinite sets S. Cantor's Theorem states that the cardinality of the power set of a set S always exceeds the cardinality of S itself. That's obvious for finite sets but far from trivial for infinite sets.

## Boltzman Entropy

A key equation for statistical mechanics formulated by Ludwig Boltzmann. It relates the entropy of a macrostate (S) to the number of microstates corresponding to that macrostate (W). A microstate describes a system by identifying the properties of each particle, this involves microscopic properties such as particle momentum and particle position. A macrostate designates collective properties of a group of particles, such as temperature, volume and pressure. The key thing here is that multiple different microstates can correspond to the same macrostate. Therefore, a simpler statement would be that the entropy is related to the arrangement of particles within the system (or the 'probability of the macrostate'). This equation can then be used to derive thermodynamic equations such as the ideal gas law.

## Cantor's Theorem

2^|S| > |s| where |S| is the cardinality of the set S. If S is a finite set then its cardinality is the number of elements in it, and things are not very interesting. But the concept of cardinality makes sense also for infinite sets. The power set of a set is the set of its subsets. It is easy to see that for finite sets S the cardinality of the power set equals 2^|S|. Thus we denote by 2^|S| the cardinality of the power set even for infinite sets S. Cantor's Theorem states that the cardinality of the power set of a set S always exceeds the cardinality of S itself. That's obvious for finite sets but far from trivial for infinite sets.

## Boltzman Entropy

A key equation for statistical mechanics formulated by Ludwig Boltzmann. It relates the entropy of a macrostate (S) to the number of microstates corresponding to that macrostate (W). A microstate describes a system by identifying the properties of each particle, this involves microscopic properties such as particle momentum and particle position. A macrostate designates collective properties of a group of particles, such as temperature, volume and pressure. The key thing here is that multiple different microstates can correspond to the same macrostate. Therefore, a simpler statement would be that the entropy is related to the arrangement of particles within the system (or the 'probability of the macrostate'). This equation can then be used to derive thermodynamic equations such as the ideal gas law.

## Integration by parts

Complicated expressions involving multiplication of two function can be integrated by as follows:

## Transport equation

Transport eqution is an equation to be regarded a fluid dynamical phenomena.

Mathematical physics is a very impressive branch of modern science. Newton and Einstein inroduced pure mathematics in the problems of physics. Mathematical physics encompasses vast range of theoretical physics topics such as theory of relativity, quantum mechanics and others. The first subject to be discussed is the vector algebra. It will not be superfluous to remark that special theory of relativity is concerned with vector algebra whereas general theory of relativity is concerned with vector calculus. One such vector calculus topic is doing calculus on manifold. The dot and cross product of two vectors in vector algebra is found using algebraic equation .

In dot product a scalar quantity is produced whereas in cross product a vector quantity is produced. Before proceding firther some elementary physics formula must be mentioned.

Translational and rotational motion are analogous. Some variables are parameters are only replaced. Examples will suffice to explain the fact.

Riemman curvature tensor has it counterpart in exterior calculus and exterior algebra.

Some equations and formulas of longitudinal wave are

scientific method, and technological tools like computers and telescopes. With the help of these tools, scientists have pieced together a lot of knowledge about space. But what do we really know about the universe, and how do we know it? Where did the universe come from? Where is it going? Did the universe have a beginning, and if so, what happened before then? What is the nature of time? Will it ever come to an end? Can we go backward in time? Recent breakthroughs in physics, made possible in part by new technology, suggest answers to some of these long-standing questions. Someday these answers may seem as obvious to us as the earth orbiting the sun—or perhaps as ridiculous as a tower of turtles. Only time (whatever that may be) will tell. 2

## Perturbation Theory

Relatively few problems in quantum mechanics have exact solutions, and thus most problems require approximations. Perturbation theory is a useful method of approximation when a problem is very similar to one that has exact solutions. The approximate results differ from the exact ones by a small correction term. Perturbation theory fails when the correction terms are not small. Consider a set of eigenfunctions and eigenvalues of a given Hamiltonian operator: ! H ˆ (0) "n (0) = En (0) "n (0) (1) Here the label n identifies a specific solution in a set and the superscript (0) denotes the “order” of approximation. The set of functions ψ form an orthonormal basis. Zeroth-order approximation means exact. Now, if we consider a first-order correction such that the true Hamiltonian is ! H ˆ = H ˆ (0) + H ˆ (1) (2). We can expect the corrected wavefunction for a state n to be of the form ! "n = "n (0) + i#n \$ ci "i (0) (3), where the summation runs over all other states i in the basis set and ci are real coefficients in the linear expansion. This implies that the corrected wavefunctions are not normalized. It can be shown that (to first order) the mixing coefficients have the values ! ci (1) = "i (0) H ˆ (1) "n (0) En (0) # Ei (0) , i \$

## peano's axiom

Peano’s three indefinables are 0, finite integer* and successor of. It is assumed, as part of the idea of succession (though it would, I think, be better to state it as a separate axiom), that every number has one and only one successor. (By successor is meant, of course, immediate successor.) Peano’s primitive propositions are then the following. (1) 0 is a number. (2) If a is a number, the successor of a is a number. (3) If two numbers have the same successor, the two numbers are identical. (4) 0 is not the successor of any number. (5) If s be a class to which belongs 0 and also the successor of every number belonging to s, then every number belongs to s. The last of these propositions is the principle of mathematical induction. 121. The mutual independence of these five propositions has been demonstrated by Peano and Padoa as follows.† (1) Giving the usual meanings to 0 and successor, but denoting by number finite integers other than 0, all the above propositions except the first are true. (2) Giving the usual meanings to 0 and successor, but denoting by number only finite integers less than 10, or less than any other specified finite integer, all the above propositions are true except the second. (3) A series which begins by an antiperiod and then becomes periodic (for example, the digits in a decimal which becomes recurring after a certain number of places) will satisfy all the above propositions except the third. (4) A periodic series (such as the hours on the clock) satisfies all except the fourth of the primitive propositions. (5) Giving to successor the meaning greater by 2, so that the successor of 0 is 2, and of 2 is 4, and so on, all the primitive propositions are satisfied except the fifth, which is not satisfied if s be the class of even numbers including 0. Thus no one of the five primitive propositions can be deduced from the other four. 122. Peano points out (loc. cit.) that other classes besides that of the finite integers satisfy the above five propositions. What he says is as follows: “There is an infinity of systems satisfying all the primitive propositions. They are all verified, e.g., by replacing number and 0 by number other than 0 and 1. All the systems which satisfy the primitive propositions have a one-one correspondence with the numbers. Number is what is obtained from all these systems by abstraction; in other words, number is the system which has all the properties enunciated in the primitive propositions, and those only.” This observation appears to me lacking in logical correctness. In the first place, the question arises: How are the various systems distinguished, which agree in satisfying the primitive propositions? How, for example, is the system beginning with

## Vector calculus

An angular velocity vector (w) and an angular acceleration vector (α). In general, these two vectors point in different directions in three-dimensional problems.

We can express w and α with their vector components.

To calculate the angular acceleration vector, we compute the difference in the angular velocity vector over a very small time step Δt, where Δt→0. To illustrate, see the figure below.

Using calculus, the angular acceleration is calculated by taking the limit as Δt→0, where

Now there is sometimes a problem in 2D plane where derivative of a vector is needed to be evaluated. This is illustrated with the following example. Let’s assume a wheel of radius R is rotating about a fixed point o, with a counterclockwise angular velocity w, as shown below. We wish to find the velocity of point P on the wheel (vp)

To solve this problem we can express the position of the point P in terms of polar coordinates (R,θ). This will allow us to fixate the velocity of point P. For the point P defined in polar coordinates (as shown below), we can find a general equation for its velocity.

We now set up the position P with respect to time as
x(t) = Rcosθ
y(t) = Rsinθ
To find the velocity, take the first derivative of x(t) and y(t) with respect to time:

Since dθ/dt = w we can write

The point P corresponds to θ = 90 so dx/dt = -Rw (pointing towards left)
We can derive this also using vector derivative approach by setting

where i and j are defined as unit vectors pointing along the positive x and y axes (respectively), at the given instant. The unit vectors i and j are also defined as fixed to the wheel and rotating with it. This implies that the point P does not move relative to the unit vectors i and j. As a result, the terms Rcosθ and Rsinθ are constant in the above two equations. To better understand this consider the following illustration which shows the position of the wheel at two consecutive instants:

And at an instant next

As you can see, between instants 1 and 2 the position of point P does not move relative to the unit vectors i and j.
We are now ready to differentiate the above two equations with respect to time:

At the given instant, the velocity at point P is given by setting θ = 90° in the above equations. This gives us

Using calculus, and taking the limit as Δt→0:

THEREFORE

As a result

## Lie groups and Lie algebra

A group is a mathematical structure equipped with certain operations. The group of integers is infinite, and so is the braid group B_n, which we discussed in Chapter 5, for each fixed n = 2, 3, 4,... (Bn consists of braids with n threads; there are infinitely many such braids). The group of rotations of a round table, which consists of all points on a circle, is also an infinite group.
But there is an important distinction between the group of integers and the circle group. The group of integers is discrete; that is to say, its elements do not combine into a continuous geometric shape in any natural sense. In contrast, we can change the angle of rotation continuously between 0 and 360 degrees. And together, these angles merge into a geometric shape: namely, the circle. Mathematicians call such shapes manifolds.
The group of integers and the braid groups belong to the family of discrete infinite groups in the Kingdom of Mathematics. And the circle group belongs to another family, that of Lie groups. Simply put, a Lie group is a group whose elements are points of a manifold. So this concept is the offspring resulting from the marriage of two mathematical concepts: group and manifold.

Group heirarchy can be depicted with a diagram.

The group of rotations of the sphere has a name in math: the special orthogonal group of the 3-dimensional space, or, as it is commonly abbreviated, SO(3). We can think of the symmetries of the sphere as transformations of the 3- dimensional space in which the sphere is embedded. These transformations are orthogonal, meaning that they preserve all distances.
The groups SO(2) and SO(3) are not only groups but also manifolds (that is, geometric shapes). The group SO(2) is the circle, which is a manifold. So SO(2) is a group and a manifold. That’s why we say that it is a Lie group.
To explain what lie algebra is , it is necessary to understand what a tangent space is. A tangent space of a circle at a point is a straight line tangent to the circle at that point. Likewise, any curve (that is, a one-dimensional manifold) can be approximated near a given point by a tangent line. Similarly, a sphere can be approximated at a given point. by a tangent plane. And an n-dimensional manifold may be approximated at a given point by a flat n-dimensional space.
Now, on any Lie group we have a special point, which is the identity element of this group. We take the tangent space to the Lie group at this point – and voilà, that is the Lie algebra of this Lie group. So each Lie group has its own Lie algebra, which is like a younger sister of the Lie group.
For example, the circle group is a Lie group, and the identity element of this group is a particular point on this circle corresponding to the angle 0. The tangent line at this point is therefore the Lie algebra of the circle group. Alas, we cannot draw a picture of the group SO(3) and its tangent space because they are both three-dimensional. But the mathematical theory describing tangent spaces is set up in such a way that it works equally well in all dimensions. If we want to ponder how things work, we can model them on one-or two-dimensional examples (like a circle or a sphere). In doing so, we use lower-dimensional manifolds as metaphors for more complicated, higher-dimensional manifolds. But we do not have to do this; the language of mathematics enables us to transcend our limited visual intuition. Mathematically, the Lie algebra of an ndimensional Lie group is an n-dimensional flat space, also known as a vector space.
Formally a Lie algebra is a vector space associated with a non-associative opearation called Lie bracket, a bilinear map [:,:] : gXg -> g satidfying Jacobi identity.

## Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures (i.e groups ) by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and its algebraic operations (for example, matrix addition, matrix multiplication). Definition: There are two ways to say what a representation is. The first uses the idea of an action, generalizing the way that matrices act on column vectors by matrix multiplication. A representation of a group G or (associative or Lie) algebra A on a vector space V is a map

That is a matrix acts on some elements v to produce another element v in the same vector space.
With the two properties. First, for any g in G (or a in A), the map

is linear over F.
φ is called the group action.

## Cauchy integral formula

Let C be a counterclockwise oriented circle living in the open set U. If the function f is holomorphic on U, then

for all points z inside the circle C. This fundamental formula was discovered by Cauchy (1789–1857) in 1831 during his exile in Turin (Italy).

## Analytic Continuation and the Local-Global Principle

Let f, g : U → R be holomorphic functions on the open, arcwise connected set U such that f(zn) = g(zn) for all n = 1, 2, . . . , and the sequence (zn) with zn /= a for all n converges to some point a in U as n → +∞. Then f = g on U.
Let U and V be open, arcwise connected sets in the complex plane such that U ⊂ V. If f : U → C and F : V → C are holomorphic and f = F on U, then F is called the holomorphic (or analytic) extension of f. This holomorphic (or analytic) continuation is uniquely determined. For example, the power series
f(Z) = 1 + z + z^2 + z^3 + .....
is convergent for all z ∈ C with |z| < 1. Since f(z) = 1 1−z , the function f can be uniquely extended to the holomorphic function F : C \ {1} → C given by

As a second example, consider the energy function

This function allows the power series expansion

for all complex energy values E with |E − 1| < 1. The unique global analytic continuation is given by the double-valued expression

We also write F(E) = √E. The fist series only converge in |E-1| < 1 whereas the final expression holds for all E belonging to C. Hence we have continued the function analytically.

## Integrals and Riemann Surfaces

Algebraic integrals: Let R = R(z,w) be a rational function with respect to the complex variables z and w (i.e., R is the quotient of two polynomials with respect to z and w). By an algebraic integral or Abelian integral, we mean an integral of the form

where C is a curve, and the function w = w(z) is given by the equation

Here, p0, p1, . . . are polynomials. Such integrals were first studied by Abel (1802– 1829). In an ingenious way, Riemann (1826–1866) discovered that algebraic integrals can be understood best by using the topological concept of the Riemann surface. Let us explain this by considering a simple example.
The Riemann surface of the logarithmic function. For each given complex number z |= 0, the equation

has a set of solutions which we denote by w = lnz. Explicitly,

This function is many-valued on the punctured complex plane C \ {0}. It was Riemann’s idea to construct a set R such that the function
ln : R -> C
is single-valued on R. This can be done easily. To this end, set S_k := C \ {0}, k= 0,±1,±2, .
We cut each sheet Sk along the negative real axis, and we glue the sheets along the cuts together in the following way.
• If we start from the point z = 1 on the sheet S0, and we move counterclockwise along the unit circle, then we change from the sheet Sk to the sheet Sk+1 at the point z = −1 where k = 0, 1, 2, . . . .
• Similarly, if we start from the point z = 1 at the sheet S0 and we move clockwise along the unit circle, then we change from Sk to Sk−1 at the point z = −1 where k = 0,−1,−2, . . .
Furthermore, if z ∈ Sk, then we define ln z := ln |z| + iargz + 2πk. The set R is called the Riemann surface of the function ln . This is an ‘infinite round staircase’. If we cut the sheet S0 along the negative real axis, then the restriction ln : S_0\ ]−∞, 0] → C
is called the principal branch of the logarithmic function; this function is singlevalued and holomorphic. Explicitly, for the principal branch,
ln z = ln|z| + iarg z, −π < argz < π.
For all complex numbers z with |z| < 1, the principal branch allows the power series expansion

Application to integrals. The concept of the Riemann surface is extremely useful for computing curve integrals. The issue is that we have to choose curves on the Riemann surface of the primitive function to the integrand.
For example, consider the smooth curve C : z = z(t), t0 ≤ t ≤ t1. By the fundamental theorem of calculus,

However, this value is not well-defined, since the logarithmic function is many-valued. This defect can be cured completely if we regard C as a curve on the Riemann surface R of the logarithmic function. For example, let C denote the counterclockwise oriented unit circle on the complex plane. Now regard C as a curve on the Riemann surface R of the logarithmic function. As an example, let us start at the point z = 1 on the sheet S0. Moving counterclockwise, we end up at the point z = 1 on the sheet S1. Hence

The same result is obtained by using Cauchy’s residue theorem. The Riemann surface R reveals the natural background of the residue theorem.

## Algebraic curves and Riemann surfaces

The real equation
w − z = 0, z,w ∈ R represents a real curve, namely, a straight line through the origin. The complex equation
w − z = 0, z,w ∈ C
represents a complex curve (also called 1-dimensional complex manifold). This complex curve corresponds to the complex plane, C, which is equivalent to the two dimensional real plane, R2. If we compactify the complex plane, C, then we obtain the closed complex plane, C, which is in one-to-one correspondence to the Riemann sphere, S2. In terms of the theory of manifolds to be introduced in Sect. 5.4 on page 234, Riemann surfaces are defined to be 1-dimensional, complex, arcwise connected manifolds ( 2 real dimension). Using this terminology, the following hold true:
The closed complex plane and the Riemann sphere are conformally equivalent compact Riemann surfaces of genus zero .
Furthermore, the real equation w2 − z = 0, z,w ∈ R describes a parabola. The complex extension w2 − z = 0, z,w ∈ C describes the Riemann surface to the function w = √z. After compactification, this is a compact Riemann surface which is conformally equivalent to the Riemann sphere. More generally, each of the algebraic equations (4.5) describes, after passing to connected components and using compactification, a compact Riemann surface of genus g where g = 0, 1, 2, . . . In 1907, Poincar´e and Koebe proved independently the famous uniformization theorem telling us that
Each algebraic curve (and hence each compact Riemann surface) possesses a global smooth parametrization z = z(t), w = w(t) for all parameters t ∈ T

where the parameter space T has to be chosen in an appropriate manner. This way, many special functions of mathematical physics appear as parametrizations of algebraic curves (trigonometric functions, elliptic functions, modular functions, and automorphic functions). For example, the complex unit circle, that is, the complex curve z^2 + w^2 − 1 = 0, z,w ∈ C possesses the global parametrization z = cost, w = sint, t ∈ C. Let e1, e2, e3 be three pairwise different complex numbers. Then, the algebraic curve w^2 − 4(z − e1)(z − e2)(z − e3) = 0, z,w ∈ C
possesses the global parametrization z = ℘(t), w = ℘(t), t ∈ C where ℘ denotes the Weiererstrass elliptic function (i.e., ℘ is double-periodic). The Riemann surface of the algebraic curve (4.7) is conformally equivalent to a torus which has always the genus g = 1. The Riemann surface to the complex curve z − ew = 0, z,w ∈ C is nothing other than the Riemann surface to the function w = lnz. The space of string states. Riemann surfaces and their applications in string theory will be studied in Volume VI. At this point let us only mention that Compact Riemann surfaces correspond to string states; conformally equivalent Riemann surfaces represent the same string state.

## Domains of Holomorphy

It turns out that there exists a crucial difference between holomorphic functions on the complex plane and holomorphic functions on the higher-dimensional complex spaces C2,C3, . . . This concerns analytic continuation. Let us discuss this. For an open set U in C^N, the function f : U → C^N is called holomorphic on U iff the partial derivatives
∂f/∂zj, j= 1, . . . , N exist on U. This is equivalent to the fact that the function f can be locally represented by a power series expansion which converges absolutely in some open neighborhood of each point in U.
An open, arcwise connected subset U of C^N is called a domain of holomorphy iff there exists a holomorphic function f : U → C which cannot be extended to a holomorphic function on a larger open, arcwise connected set.
On the complex plane C, each open, arcwise connected set U is a domain of holomorphy. This theorem does not remain true for C2,C3, . . . For example, consider the set

with

Then, each holomorphic function f : W → C can be extended to a holomorphic function f :U →C on the open unit ball

but a further extension to a larger open set is not always possible. This tells us that the open set W is not a domain of holomorphy. In contrast to this, the open ball U is a domain of holomorphy.
Each convex open subset of CN is a domain of holomorphy
The famous 1938 Bochner theorem on the analytic continuation of functions of several variables tells us the following. Consider a so-called tube

where Ω is a convex open subset of RN, N = 1, 2, . . .

### Reference materials:

Law of thermodynamics
A briefer history of time by S. Hawking
A brief history of time by S. Hawking
Quantum mechanics
Grand Design by Stephen Hawking
perihelion of mercury by Feynman