## complex analysis churchill

Complex numbers is formed with a real and imaginary number.

Complex number is defined on argon plane. The abscissa is the real number line and the ordinate is the imaginary number (squreroot(-1)) line. So a point on the argon plane can be represented as a+ib where a is the number of units that represent the abscissa and b is the number of imaginary units i. Argoon plane or complex plane looks like this :

The use of imaginary number is pervasive in physics and engineering. It simplifies a lot of calculation. Electricity , for example , can ba modeled with imaginary quantity. In quantum field theory all the integrations are computed using this imaginary number. The algebra of complex numbers is different. It is more like vector algebra. For example multiplication of two complex numbers would form a complex number with different amplitude and different angle.

Other operations of complex numbers can be stated in a single package :

A typical quantum field is repsented by this formula

An important theorem in complex analysis is the Demoiver theorem which relates cosine and sine with the imaginary quantity i.

So any complex number can be represented using sine and cosine in this way

Suppose R is a set of all sets that are not members of themselves. Now is this set R is a member of itself or not? If it is a member of itself then it is one of the sets which are not members of themseves. So it is not member of itself. And if this set R is not member of itself then by definition it is a member of itself. So in either way we come to a contradiction!
Nth root of unity is expressed using imaginary number:

So all solutions when raised to power n give unity or 1.

## Complex numbers identity

There are many remarkable identities involving complex numbers :

## Cauchy Riemann equation

The Cauchy Riemann equation can be derived using usual argument of limit :

Suppose R is a set of all sets that are not members of themselves. Now is this set R is a member of itself or not? If it is a member of itself then it is one of the sets which are not members of themseves. So it is not member of itself. And if this set R is not member of itself then by definition it is a member of itself. So in either way we come to a contradiction!

Suppose R is a set of all sets that are not members of themselves. Now is this set R is a member of itself or not? If it is a member of itself then it is one of the sets which are not members of themseves. So it is not member of itself. And if this set R is not member of itself then by definition it is a member of itself. So in either way we come to a contradiction!

The result is :

## Green theorem

Green theorem establishes an important result in complex calculus. Green's theorem gives a relationship between the line integral of a two-dimensional vector field over a closed path in the plane and the double integral over the region it encloses. The fact that the integral of a (two-dimensional) conservative field over a closed path is zero is a special case of Green's theorem.

## Complex inversion formula

It is the same thing as the Laplace transform which tranform a function into complex domain.

## Cauchy integral formula

It is one of most remarkable theorems in complex analysis. If a function is analytic in domain R then cauchy's integral formula tells :

That is if you know the values of the function f(z) along a contour C you can determine its value in anywhere inside the contour. And you can determine its derivative also.

This brings us to contour integration.

## Contour integration

Contour integal is defined along a contour or closed loop in complex plane. That is the range of the values of the function f(z) falls on the contour curve. An example can make it clear

In evaluating those integral we have used estimation lemma which states as follows:
In mathematics the estimation lemma, also known as the ML inequality, gives an upper bound for a contour integral. If f is a complex-valued, continuous function on the contour Γ and if its absolute value |f(z)| is bounded by a constant M for all z on Γ, then

When calculating contour integral a branch cut might fall inside the contour. In that case the contour needs to be distorted to evade the branch cut. A branch cut is a curve (with ends possibly open, closed, or half-open) in the complex plane across which an analytic multivalued function is discontinuous. For convenience, branch cuts are often taken as lines or line segments. So in case of branch cuts the integral becomes troublesome. A roundabout way is applied

The integation done above is a kind of improper integral. Improper integral is the integral which gives infinite value or beomes divergent at the limit point. So a dummy variable η is used here. It is infinitesimal quantity which tends towards zero. More can be said about improper integral , which at the moment best left. The most notable example of this kind of improper integral can be seen in Feynman's propagator's derivation. Another example of improper integral is the integral of 1/x :

## More on contour integration

Firstly a hologous deformation is the kind of deformation in which one path is continuously deformed into other path. Two paths that are deformable into one other are said to be in the same homology class. With a homologous deformation, it is legitimate for parts of paths to cancel one another out, provided that those portions are being traversed in opposite directions. By contrast homotopy class does not permit this kind of cancellation. Two paths that are deformable one into the other in this way are said to belong to the same homology class. By contrast, homotopic deformations do not permit this kind of cancellation. Paths deformable one into another, where such cancellation are not permitted, belong to the same homotopy class. Homotopic curves are always homologous, but not necessarily the other way around. Both homotopy and homology are to do with equivalence under continuous motions. Thus they are part of the same topology.

subject of topology. We shall be seeing different aspects of topology playing important roles in other areas later.
The function f (z) = 1/z is in fact one for which diVerent answers are obtained when the paths are not homologous. We can see why this must be so from what we already know about logarithms. Towards the end of the previous chapter, it was noted that log z is an indefinite integral of 1/z. (In fact, this was only stated for a real variable x, but the same reasoning that obtains the real answer will also obtain the corresponding complex answer. This is a general principle, applying to our other explicit formulae also.) We therefore have

But recall, from (multi-valuedness of logarithm), that there are different alternative ‘answers’ to a complex logarithm. More to the point is that we can get continuously from one answer to another. To illustrate this, let us keep a fixed and allow b to vary. In fact, we are going to allow b to circle continuously once around the origin in a positive (i.e. anticlockwise) sense , restoring it to its original position. Remember, , that the imaginary part of log b is simply its argument (i.e. the angle that b makes with the positive real axis, measured in the positive sense;). This argument increases precisely by 2p in the course of this motion, so we Wnd that log b has increased by 2pi (). Thus, the value of our integral is increased by 2pi when the path over which the integral is performed winds once more (in the positive sense) about the origin. We can rephrase this result in terms of closed contours, the existence of which is a characteristic and powerful feature of complex analysis. Let us consider the diVerence between the second and the Wrst of our two paths, that is to say, we traverse the second path Wrst and then we traverse the Wrst path in the reverse direction . We consider this diVerence in the homologous sense, so we can cancel out portions that 'double back' and straighten out the rest, in a continuous fashion. The result is a closed

path—or contour—that loops just once about the origin , and it is not concerned with the location of either a or b. This gives an example of a (closed) contour integral, usually written with the symbol ∫ and we Wnd, in this example,

Of course, when using this symbol, we must be careful to make clear which actual contour is being used—or, rather, which homology class of contour is being used. If our contour had wound around twice (in the positive sense), then we would get the answer 4pi. If it had wound once around the origin in the opposite direction (i.e. clockwise), then the answer would have been -2π.

## Z-transform

Z-transform is often used in digital filter designing. It is a method which converts a discrete-time-signal which is a sequence of real or complex numbers into complex frequecy domain representation.

Where a is an integer and z is a complex number of the form

## Extended Cauchy's theorem

An example of evaluating integral using Cauchy's theorem:

## Some theorems proved by Euler

The fourth equation is the famous equation related to graph theory. E represents the number of edges , F represents the number of faces and V represents the number of vertices. When this is the case then the equation always holds true.

## Maclaurin series

Maclaurin series is the special case of Taylor's series where we evaluate the function and its derivative at the origin 0. Here is the example:

I was once thinking how calculator calculate the values of trigonometric functions like sines, cosines and tangent . It is possible that algorithm is developed to evaluate the Maclaurin series. This way it is easy to calculate those.

## Laurent series and convergence

In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. It has the following mathematical expression:

The red colored area is the annulus where the function f(z) is analytic.

## Gamma Function

Gamma function is a special type function . It can be represented with complex number as follows:

## Simply connected domain

A picture is worth a thousand words.
A simply connected domain or region is any path connected domain where one can shrink any closed path or loop into a point by continuous deformation. When this in not possible in any region or domain in complex plane then the region is multiply-connected region.

With this definition of simply connected region we proceed to analytic continuation of holomorphic function.
A holomorphic function is the function which is analytic in the domain defined through it. By analytic it is to mean that the function is complex differentiable. Analytic continuation is a process by which we extend the domain of a holomorphic function , that we started with initially. Suppose f is an analytic function defined on a non-empty open subset U of the complex plane C. If V is a larger open subset of C, containing U, and F is an analytic function defined on V such that
F(z) = f(z) for all z in U,
then F is called an analytic continuation of f. In other words, the restriction of F to U is the function f we started with. If there is an analytic continuation it will be unique.
The complex logarithm can be analytically continued with the addition of 2π to the function every time we circle around the complex plane.

This special surface is called principle value of Riemann surface. It has some other special properties. The above plot is based on the complex logarithm. Complex logarithm is found by taking logarithm of any complex number. This is the actual derivation:

It is a multi-valued function , which is to say that, given a single argument it has many values. It also has a branch point at z=0 Where it jumps. The complex part of it is the imaginary number which is i times argument of complex number z. The argument can be written as θ + multiples of 360(2π). Riemannn used this property and thought about a more generalized domain of such function. By repeatedly stacking one such surface over we can construct Riemann surface. It looks like a spiral ramp flattened down vertically to the complex-plane. In this extended domain the complex logarithm is single-valued as after each revolution 2πi gets added to the function.

## Analytic continuation

Analytic continuation is a method in complex analysis to extend the domain of a given analytic function. Suppose f is an analytic function defined on an open subset U in complex plane C. If V is larger set which includes U as a subset then F is the analytic continuation of f defined on V such that f(x) = F(Z) for all z in U.
Analytic continuation is often very unique. Suppose two function F1 and F2 are the analytic continuation of the same function f defined on U. F1 and F2 are defined on the connected domain V . Then for all z in U
F1(z) = F2(z) = f(z) and so F1 = F2 for all z in V . This is because F1 - F2 vanishes in the domain U hence it will vanish in the entire domain V. F1 and F2 are unique.
Now we will demonstrate an example of analytic continuation.
With a particular analytic function f given by its power series expansion centered at point z = 1

its radius of convergence is 1. That is f is analytic on the open set U = { |z-1| < 1 } which has a boundary dD = { |z-1| = 1 } . In fact the serie diverges at the point z= 0 of dD . Suppose we do not know f(z) = 1/z and we expand the function f for all a ε U in the power series.

We will now calculate a(k) and determine if whether this new power series converges in the open region V which is not contained in U. If that is so, we have the analytic continuation of f in V .
The distance from a to dU is &rau; = 1 - |1-a| > 0. Take 0 < r < &rau; . Let D be the disc of radius r around a. And let dD be its boundary. Then D U dD ε U. The coefficients a(k) are

So

Which has a radius of convergence a and V = { |z-a| < |a| }. If we choose a ε U and |a| > 1 , then V will not be a subset of U. V will be larger in area than U. The plot shows that a = 1/3 (1 + 3i).

## Real Analysis

Real analyis is the study of real numbers and sequences and series of various types. It also studies the properties of real number like maximum, infinimum , supremum and rationality.

Sequence is a collection of mathematical objects like numbers or functions. The set of real numbers or natural numbers can be called a sequence. In sequence repetitions are allowed. In every sequence there is definite order between the elements. Sequence of continuously differentiable functions has some specific properties as follows:

## Hypercomplex Algebra

Hypercomplex algebra is the extension of usual two real dimensional complex algebra. It is a generalized form of complex algebra of higher order. The rules of addition and multiplication of higher dimensional complex numbers is different than the rules of usual complex algebra. Higher order complex numbers are called hypercomplex numbers. The renowned Irish mathematician William Rowan Hamilton (1805–1865) was one who puzzled long and deeply over this matter. Eventually, on the 16 October 1843, while on a walk with his wife along the Royal Canal in Dublin, the answer came to him, and he was so excited by this discovery that he immediately carved his fundamental equations

on a stone of Dublin’s Brougham Bridge. Each of the three quantities i, j, and k is an independent ‘square root of -1’ (like the single i of complex numbers) and the general combination

where t, u, v, and w are real numbers, defines the general quaternion. These quantities satisfy all the normal laws of algebra bar one. The exception— and this was the true novelty1 of Hamilton’s entities—was the violation of the commutative law of multiplication. Hamilton found that
ij = -ji, jk = -kj and ki = -ik
which is in gross violation of the standard commutative law: ab = ba. Quaternions (t, form a 4-dimensional vector space over the reals, because there are just four independent ‘basis’ quantities 1, i, j, k that span the entire space of quaternions; that is, any quaternion can be expressed uniquely as a sum of real multiples of these basis elements. We shall be seeing many other examples of vector spaces later.
Quaternions also provide us with an example of what is called an algebra over the real numbers, because of the existence of a multiplication law, as described above. But what is remarkable about Hamilton’s quaternions is that, in addition, we have an operation of division or, what amounts to the same thing, a (multiplicative) inverse q^[-1] for each nonzero quaternion q. This inverse satisfies
(q^-1)(q) = 1 = (q)(q^-1)
giving the quaternions the structure of what is called a division ring, the inverse being explicitly

I will finish with the geometry of quarternion regarding hypercomplex numbers
Think of the basic quaternionic quantities i, j, k as referring to three mutually perpendicular (right-handed) axes in ordinary Euclidean 3- space. Now, we recall from §5.1 that the quantity i in ordinary complex-number theory can be interpreted in terms of the operation ‘multiply by i’ which, in its action on the complex plane, means ‘rotate through a right angle about the origin, in the positive sense’. We might imagine that we could interpret the quaternion i in the same kind of way, but now as a rotation in 3 dimensions, in the positive sense (i.e. righthanded) about the i-axis (so the (j, k)-plane plays the role of the complex plane), where we would correspondingly think of j as representing a rotation (in the positive sense) about the j-axis, and k a rotation about the k-axis

If we rotate a book through a right angle (in the right-handed sense) about i and then rotate it (in the right-handed sense) about j, we find that it ends up in a configuration (with its back spine upwards) that cannot be restored to its original state by any single rotation about k.

What we have to do to make things work is to rotate about two right angles (i.e. through 180, or π). This seems an odd thing to do, as it is certainly not a direct analogy of the way that we understood the action of the complex number i. The main trouble would seem to be that if we apply this operation twice about the same axis, we get a rotation through 360 (or 2π), which simply restores the object (say our book) back to its original state, apparently representing i^2 = 1, rather than i^2 = -1. But here is where a wonderful new idea comes in. It is an idea of considerable subtlety and importance—a mathematical importance that is fundamental to the quantum physics of basic particles such as electrons, protons, and neutrons. Ordinary solid matter could not exist without its consequences. The essential mathematical notion is that of a spinor . Spinor changes to negative when it undergoes a complete rotation of 2π Using the book's rotation it is apparent that ij = k which Hamilton equation dictates.

## Roots of unity

There are solutions of an equation , which are called nth roots of unity. These roots are complex numbers , which means thery are imaginary solutions of the same equation.

## Equiangular Spiral

Each time a rotation happens 2π is added to the argument.

## Geometry of elementary function

The squaring function z->z^2 will be something like

Path covering lemma states that if γ : [a, b] -> G is a continuous path from interval [a,b] to an open subset G of C then there is a number p>0 and a subdivision of the interval a=t0 < t1 < t2 < t3 < tn =b so that the following conditions hold.

## Gauss's fundamental theorem of algebra

"Finally, I note that it is not at all impossible that the proof, which I have based on geometric principles here, be given in a purely analytic form; but I believed the presentation which I developed here to be less abstract and to expose better the essence of the proof than one could expect from an analytic proof" --- C F Gauss
Complex numbers were introduced by Bombelli in 1550; he used complex numbers systematically in order to solve algebraic equations of third degree. The imaginary unit, i, solves the equation z^2 + 1 = 0, and we have the factorization
Z^2 + 1 = (z+i) ( z- i).
Let n = 0, 1, 2, 3 .... then for each polynomial

with complex coefficients a0, a1, . . . , there exist complex numbers z1, . . . , zn such that

For all z belonging to C.
Gauss proved this theorem for the first time in his Ph.D. thesis7 in 1799. He reduced this problem to the intersection problem for two real algebraic curves. To explain the basic idea, consider the equation z^2 + 1 = 0 FOR ALL Z BELINGING TO C.
let z = x + iy we get x ^2 - y^2 + 2ixy + 1 = 0 . Now this problem is equivalent to a system of equations
x^2 - y^2 = -1 and 2xy = 0 In terms of geometry, this system describes the intersection of a hyperbola with the y-axis . In the general case, Gauss had to study the intersection problem for two algebraic curves of nth degree. . Let us give an elegant proof of the fundamental theorem of algebra by using the winding number.
Existence of at least one zero. Set f(z) := z^n, and p(z) = f(z) + g(z). Choose a counterclockwise oriented circle C of radius R centered at the origin. Then |f(z)| = R^n on C. Furthermore

If we choose the radius R sufficiently large, then the perturbation property (P) of the winding number above tells us that
w(p) = w(f + g) = w(f) = n.
By the existence principle (E), p(z1) = 0 for some z1.
• Step 2: Factorization. By the Taylor theorem,

Hence p(z) = (z −z1)q(z) where q is a polynomial of degree n−1. By induction, we obtain the claim.
The geometric idea behind the argument from step 1 is quite easy. The map z → zn sends the circle C to a circle which winds n times around the origin. If the radius R is sufficiently large, then the map z → p(z) sends the circle C to a curve which winds n times around the origin as well, by a perturbation argument. By continuity, there must exist some point z1 inside the circle C which is mapped to the origin, that is, p(z1) = 0. The point z1 is the desired zero of the given polynomial p.
Gauss implicitly used this intuitive argument, but he had to argue in a more sophisticated way, since he did not have the rigorous theory of the winding number at hand.

## Compactification of complex plane

Each bounded sequence (zn) in the complex plane contains a convergent subsequence. This important property is not always true for unbounded sequences. In order to cure this defect, we add the symbol ∞ to the complex plane C. The set

is called the closed complex plane.
The Riemann sphere. The unit sphere S^2 := {(x, y, ζ) ∈ R3 : x^2 + y^2 + ζ^2 = 1} in the 3-dimensional Euclidean space is called the Riemann sphere. Naturally enough, the point N := (0, 0, 1) is called the North Pole of S2.

Using the figure above we set χ(z) := P. The map χ : C → S2 \ {N} can be extended to a map χ : C → S^2 by setting χ(∞) := N. A sequence (zn) in the closed complex plane is said to converge to the point z iff this is true for the corresponding points on the Riemann sphere. We now get the desired convergence theorem:
Each sequence in the closed complex plane has a convergent subsequence.
We call the closed complex plane a compactification of the complex plane. The Riemann sphere S2 is a compact topological space, and the closed complex plane C is homeomorphic to the Riemann sphere. The inverse map χ−1 : S2 \N → C is called stereographic projection. By definition, a function f is called locally holomorphic at the point z = ∞ iff the function g(z) := f (1/z). Analyticity is locally holomorphic at the point z = 0. For example, the functions f(z) := 1/z^n, n= 1, 2, . . . are locally holomorphic at the point z = ∞.
We come up with a theorem which states :
"Each holomorphic function from the closed complex plane into itself is constant."

## Mapping degree and intersection number

Sign function is a real-valued function defined as

If a function f has only a finite number of zeroes x1, x2, x3 then the degree of the function is

Instersection number is closely related to the degree of a function .

## Mathematics is the subject based on defintions, axioms and deductions....

We are given a continuous function f : [0, 1] → R which does not vanish on the boundary of the interval. If deg(f) |= 0, then the function f has a zero.
The most important elementary geometric notion is the concept of intersection number

The concept of mapping degree of a function can be extended to mapping degree as

Local mapping degree: Consider maps as pictured in Fig. 5.7. If f(P) = P` and f preserves (resp. reverses) orientation in a sufficiently small neighborhood of the point P, then we set
degP f := 1 (resp. degP f = −1). • Global mapping degree: If a map f : A → B globally preserves (resp. reverses) orientation, then we set deg f := 1 (resp. deg f = −1).

## Riemannian geometry of two dimensional sphere

To begin with, let us introduce spherical coordinates
−π < ϕ ≤ π,
−π/2 ≤ ϑ ≤ π/2 .
Here, ϕ and ϑ denote geographic longitude and geographic latitude, respectively. Moreover, we get the following:
• equator: ϑ = 0;
• North Pole: ϑ = π/2 ;
• South Pole: ϑ = −π/2 ;
• meridian: ϕ = const;
• parallel of latitude: ϑ = const.
In terms of Cartesian coordinates x, y, z, the sphere S^2 R can be parametrized in the following way:

In fact, it follows from cos^2 α + sin^2 α = 1 that x^2 + y^2 + z^2 = R2. Now consider a smooth curve

## Complex numbers in quantum theory

If an observer measures the state ψ by a measurement device, then this corresponds to the decomposition

where ϕ1, . . ., ϕN is a basis of the Hilbert space X, and c1, . . . , cN are complex numbers. In order to get a physical interpretation of this decomposition, assume additionally that the basis ϕ1, . . . , ϕN forms an orthonormal system, that is, <ϕj |ϕk> = δjk for all j, k.14 Then

Finally, assume that the state ψ is normalized, that is, <ψ|ψ> = 1. Then

### 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
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