The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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Conformal Mapping from Equilateral triangle to Isosceles Right Triangle

This is an exercise problem. Does there exist a conformal mapping from an equilateral triangle onto an isosceles right triangle such that, under correspondence of boundary, vertices are mapped to ...
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1answer
10 views

Laurent series and residues $f(z^2)$

If $f(z)$ is analytic in $ 0 < |z| < n$, what is the residue of the function $f(z^2)$ at $z = 0$? Attempt If $f(z)$ has a pole of order n at $z=0$ it seems like the residue of $f(z^2)$ would ...
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6 views

Invariant point (flex) invariant under projective transformations.

Let $C$ be a projective curve in $\mathbb{P}_2$ defined by a homogeneous polynomial $P(x, y, z)$ and let $\alpha$ be a linear transformation of $\mathbb{C}^3$. Let $Q$ be the homogeneous polynomial $Q ...
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1answer
33 views

Number of Zeros of an equation

Why is it that the equation $z^5=0$ has five zeros, seeing as $z=0$ is the only solution? (When $z$ is a complex number) The context for this is Rouche's theorem.
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32 views

A holomorphic function which takes real values at $ 1/n $ has real coefficients

I'm facing the following problem: Let $ f : U \rightarrow \mathbb{C} $ be holomorphic ($ U $ is a complex domain and $ 0 \in U $). Suppose that for all $ n = 1,2,3 \dots $ it holds that $ f(1/n) \in ...
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3answers
36 views

$\langle A,B\rangle = \operatorname{tr}(B^*A)$

"define the inner product of two matrices $A$ and $B$ in $M_{n\times n}(F)$ by $$\langle A,B \rangle = \operatorname{tr}(B^*A), $$ where the {conjugate transpose} (or {adjoint}) $B^*$ of a matrix $B$ ...
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1answer
15 views

Laplace transform involving the gamma function.

Does anyone know how to evaluate the following integral $$ \int_{0}^{\infty} \frac{e^{-qs}\alpha^{s}}{\Gamma(s)\Gamma(s)}\text{d}s $$ where $q,\,\alpha > 0$? I've done some digging in usual ...
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2answers
49 views

Computing $\int_{-\infty}^{\infty}\frac{\cos(x)}{x^2 + x + 2} \, dx$

I wish to compute $$\int_{-\infty}^{\infty}\frac{\cos(x)}{x^2 + x + 2} \, dx$$ The singularities are $\pm \frac{\sqrt{7}}{2}i - \frac{1}{2}$ I then make a half-circle contour $C = \{Re^{it}, t \in ...
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2answers
33 views

Residue $\frac{e^z}{z^3\sin(z)}$

I want to find the residue of $$\frac{e^z}{z^3\sin(z)}$$ and get $$ \frac 1 {3!} \lim_{z \to 0} \left( \frac{d^3}{dz^3} \left(\frac{ze^z}{\sin(z)} \right)\right) = \frac{1}{3}$$ Can anyone confirm ...
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3answers
45 views

Entire functions with a given condition.

Find all entire functions satisfying the condition that $f(2z)=f(z)^{2}$ and $f(0)\ne 0.$
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30 views

Ideal of Ring of holomorphic functions

Can you tell me a non trivial ideal of ring of holomorphic functions from C to C.
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0answers
52 views

Compute $\int_{0}^{\infty}\frac{x\sin(x)}{(x^2 + a^2)(x^2 + b^2)}dx$

I wish to compute $\int_{0}^{\infty}\frac{x\sin(x)}{(x^2 + a^2)(x^2 + b^2)}dx$ using complex analysis, but if that isn't possible using just regular calculus is fine. $a>0, a \neq b$ I was ...
2
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1answer
17 views

Link between harmonic and holomorphic functions on a non-simply connected domain.

There is a theorem that states that if a function $h$ is harmonic on a simply connected domain, there exists a holomorphic function $f$ such that $h = Re f$. Now, I am having a problem with the ...
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1answer
24 views

Error term of a Tauberian theorem and lattice points in circles

Suppose $\{a_n\}$ is a sequence of non-negative real numbers, $a_n = O(n^M)$ for a positive number $M$ and it's Dirichlet series $L(s)=\sum \frac{a_n}{n^s}$ has an analytic continuation to a ...
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0answers
43 views

Problem with statement of Needham's Visual Complex analysis exercise 15 chapter 3 page 185

The exercise 15 and accompanying diagram above show 3 clockwise points $q, r, s$ on a circle and a 4th point $z$ on the circle between $q$ and $s$, and a second diagram with the same three points ...
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6answers
67 views

Computing $\int_{0}^{\infty} \frac{x^2}{(x^2 + a^2)^2}dx$

I want to compute $$\int_{0}^{\infty} \frac{x^2}{(x^2 + a^2)^2}dx$$ and have tried applying trig substution with $x = a\tan(t)$, but things get a bit messy at the very end. I get $$ \left [ ...
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3answers
61 views

Computing $\int_{0}^{\infty} \frac{x^2 + a}{x^6 + a^3}dx$

I wish to compute $$\int_{0}^{\infty} \frac{x^2 + a}{x^6 + a^3}dx, \quad a>0$$ but have no contour to work with. Does anyone have ideas on how to compute this integral?
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1answer
31 views

Injectivity of analytic $f: D(0, r) \mapsto \mathbb{C}$ in the range $D(0, \min\{|f(z)|: |z| = r\})$

If $f$ is analytic on $\{|z| \leq r\}$ and $f(0) = 0, f'(z) \neq 0$. Let $\rho = \min\{|f(z)|: |z|=r\}$. Is it true that if $|w| \leq \rho$ in the image of $f$, then $f(z) = w$ has only one zero on ...
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2answers
21 views

Laurent Series of $(z^2 + 3z + 2)e^{\frac{1}{z+1}}$

I want to find the Laurent series of $(z^2 + 3z + 2)e^{\frac{1}{z+1}}$ around $z_0 = -1$. However, since this is not a fraction in the form $\frac{a}{z-b}$, I am not sure how to calculate it.
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1answer
23 views

Generalised Cauchy Integral formula

Is the step I have highlighted in the proof incorrect? We only know $f$ is holomorphic for $|z|<1$, so why are we able to use the generalised Cauchy's formula on the curve |$z|=1$?
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21 views

Conformal isomorphisms from disc to half-disc fixing 1 and -1

I am preparing for a complex analysis qual and ran into this problem on an old exam. Find all conformal isomorphisms from the unit disk $\mathbf{D}=\{z\in\mathbf{C} : |z| < 1\}$ to the semi-disk ...
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1answer
26 views

Finite number of zeros if $f$ is analytic and satisfies homologous condition

I was reading Conway's complex analysis and I encounter the following exercise. I appreciate if someone can help me. Let $G$ be open and suppose that $\gamma$ is closed rectifiable curve in $G$ ...
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4answers
120 views

Complex number, series

Show that $$\frac{1}{z^2}=1+\sum_{n=1}^\infty (n+1)(z+1)^n$$ when $|z+1|<1$ I'm having problems to resolve this type of exercise since my book has virtually no exercises of this type, these ...
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1answer
20 views

Solve polynomial equation in $\mathbb{C}[x]$

Find the polynomials $f,g \in \mathbb{C}[x]$ with complex coefficients such that: $$f(f(x))-g(g(x))=1+i,\\f(g(x))-g(f(x))=1-i$$ for all $x\in\mathbb{C}$. I think I have this problem almost ...
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1answer
27 views

Complex number, series representation

Show that for any finite value of $z$ $$e^z=e+e\sum_{n=1}^\infty \frac{(z-1)^n}{n!}$$ For $z=1$ $$f(z)=f(z_0)+\sum f^{(n)}(z_0)\frac{(z-z_0)^n}{n!}$$ equality is checked, but I do not know how to ...
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1answer
27 views

Compute $\frac{f(i\frac{\pi}{2})}{f(i\pi)}$ for an analytic function.

I came across this problem and I'm having a little trouble. Let $f$ : $\mathbb{C} \rightarrow \mathbb{C}$ be a nonconstant analytic function. Assume that on $\mathbb{R}$ the function is real-valued ...
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35 views

Explaining Cauchy's Theorem

I need to explain why the cauchy theorem does not apply to this integral $$\int_\gamma \bar{z}\ dz,$$ where $\gamma$ is the segment straight line [0,2+i] I found that the integral equals $5/2$. I ...
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23 views

Verifying a Möbius transformation

This is the formula given for a transformation that sends (z2, z3, z4) to (1, 0, $inf$).. http://imgur.com/BmNpaiU 2 I reversed z2 and z3 to keep it in this form, but had no luck at all. I also ...
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1answer
31 views

Applying Cauchy's theorem

Why is the part highlighted in green equal to zero?
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23 views

Prove $SU(2)$ is isomorphic to the group of quaternions of norm 1

How could I start finding the isomorphism? Intuitively, a quaternion can be expressed as two complex numbers $a+bi+cj+dk=a+bi+(c+di)j$, and as an element of $SU(2)$ is $\left[ \begin{array}{ c c } ...
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1answer
14 views

Complex Vector spaces inner product superposition axiom

In my studies of Quantum mechanics, the following problem with complex vector spaces has come up, specifically as regards the inner product in such a space. Now in Shankars "Principles of Quantum ...
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13 views

Complex Differentiability and its difference to real differentiability

I am currently studying a course on complex analysis and complex differentiability is defined as: $f : U \rightarrow \mathbb{C}$, where $U$ is a domain, is complex differentiable if and only if it is ...
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3answers
177 views

Does $\sqrt{i + \sqrt{i+ \sqrt{i + \sqrt{i + \cdots}}}}$ have a closed form?

I've been brushing up on my complex analysis recently, and I've come across a problem that's stumped me: What are the real and imaginary parts of $$\sqrt{i+\sqrt{i+\sqrt{i+\sqrt{i+\cdots}}}} ?$$ I ...
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1answer
42 views

Analytic inverse of $f(z) \neq 0, f(0) = 0, f'(z) \neq 0 $ within minimum modulus on boundary.

Suppose $f(z)$ is analytic on closed disk of radius $r$ and $f(0)=0$, $f'(z) \neq 0$. Show that $f$ has an analytic inverse on $\{|z| \leq m\}$ where $m$ is the minimum of $|f(z)|$ on $\{|z| = r\}$. ...
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Question about a challenging problem in complex analysis, any ideas are welcome [on hold]

Is there any body who have any idea to solve this problem? It seems that it is a challenging problem. The problem is as follows:
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32 views

True or False: If f is analytic and maps the deleted unit disk to the unit disk, then 0 is not a pole for f.

I am studying for my final exam in complex variables and I ran across this true or false question. True or False: If $f:D(0;1)\setminus\{0\} \rightarrow D(0;1)$ is analytic, then $0$ is not a pole ...
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1answer
21 views

Theorem regarding primitives and complex integration

In order to apply this theorem must $f$ be continuous in the entirety of the open set $\Omega$ or only on the curve $\gamma$?
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28 views

Complex singularity exponent

I am studying about complex singularity exponents of holomorphic functions. I need some help to clarify a few things: First, what a complex singularity exponent is, for the holomorphic function ...
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2answers
55 views

Computing the complex integral?

I am dealing with the following: $$\int_{0}^{\infty}\frac{x\sin(x)}{(x^2+a^2)(x^2+b^2)}dx$$ Furthermore, I know $a,b>0$ and I know $a\neq b$. I believe this is using Jordan's Lemma? I see that the ...
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Integrating $\int_0^1\cos(\lambda x^3)dx$ using the saddle point method [on hold]

Find the leading term of asymptotics as $\lambda\to\infty$ $I(\lambda)=\int_0^1\cos(\lambda x^3)dx$ Using method of saddle points along a certain contour. I am having trouble approaching this ...
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19 views

How to determine contours by looking at the exponential integrands?

I know that we determine the contours in contour integrals by looking at the exponential integrand (assuming there is indeed an exponential integrand in the given integral) but I don't know how. For ...
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32 views

How are singularities of complex functions classified?

I have the function $$\frac{z^2+1}{z^3+6z^2+z}$$ And I wish to find the residues at $z_0=0$ and $z_0=2\sqrt{2}-3$ because they are within my given contour. However, I am really confused when it comes ...
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2answers
28 views

Have I Correctly Defined the Set of Nonzero Complex Numbers $\mathbb{C^*}$?

If the set of complex numbers $\mathbb{C} = \{a+bi\mid a,b \in \mathbb{R}\}$, then what would be the definition of the set of nonzero complex numbers? Am I right in defining such a set as ...
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12 views

Is the Möbius inversion applicable in the case of number functions with values in $Q(x)$

I am looking for the cause of an erroneous calculation I did the details I cant present here. I guess a "Möbius inversion" I apply might be the cause. Normally the Möbius inversion is valid for ...
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2answers
62 views

Image of curves in the complex plane

I'm not really sure what I'm being asked in this question. If $x=C,y=C$ doesn't that mean $z=C+iC$?
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1answer
15 views

Showing a complex function is nowhere differentiable in a certain disc

I have a function and I am asked to prove that it is nowhere differentiable on an open disc. I found the cauchy riemann equations and saw that is is satisfied at the origin. I don't know what to do ...
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32 views

Counting zeros of an analytic function [on hold]

Suppose f is analytic on $Ball_R(0)$ and satisfies $|f (z)| < R$ for $| z| = R$. Using Complex analytic methods (such as Rouche's theorem), how can I find the number of solutions (counting ...
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1answer
37 views

Help With Bromwich Inversion Formula Proof

To prove(copied from handwritten notes so possibly wrong): Bromwich Inversion Formula. Fix $x_0∈ℝ $. If $F$ is complex analytic on $\{z:\Re z > x_0\} $ and for every $x>x_0$, $y↦ F(x + iy )$ ...
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2answers
42 views

Under what conditions do you use that $\operatorname{Res}{(f(z)/g(z))}=f(z_0)/g'(z_0)$?

In complex analysis, this seems to be a really helpful way to avoid having to expand out Laurent series. I am unclear, however, when it is appropriate to use this property. In specific, I'm worried I ...
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0answers
15 views

Laurent series expansion for powers of n?

I wish to expand the function: $$\dfrac{e^z}{z^n-c^n}$$ about the point $z_0=c$, where c is a constant greater than 0 and n is greater than 2. So I have that $e^{z-c}$ expands to ...