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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Using Residue Theorem/Laurent Series to evaluate $\int_{c}$ $\frac{z^m}{2-\frac{1}{Z}} dz$

Let $m$ be an integer and $C$ be the circle $C(0; 1)$ traversed in the counterclockwise direction. What is the value of $\int_{c}$ $\frac{z^m}{2-\frac{1}{Z}} dz$ a) when $m\ge-1$? b) when ...
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163 views

Is it true that $log(i) = \frac\pi2i$ ? If so, are both of these legitimate proofs? They seem too beautiful not to be…

Sorry if this is a naive question. I have not yet taken any upper level math courses involving complex numbers. However, in preparation for those courses, together with utilizing the knowledge that ...
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32 views

The existence of a complex continuation of the logarithm

in the book The Prime Numbers and Their Distribution from Tenenbaum is a note about the existence of a complex continuation of the logarithm: Let $\alpha>0$ and an analytic function $f(s)$ with ...
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11 views

How to interpret $\Phi(z) \sim c_{\kappa}z^{\kappa}+\mathcal{O}(z^{\kappa-1})$

In my reading, I've come across the following statement: The function $\Phi(z)$ is said to have degree $\kappa$ at infinity if $\Phi(z) \sim c_{\kappa}z^{\kappa}+\mathcal{O}(z^{\kappa-1})$ as ...
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48 views

How to integrate $e^{-\pi x^2} \cos(2\pi x w)$

How to evaluate the following integral? The answer is $e^{-\pi w^2}$ but I don't know how do we get it. $$\int_{-\infty}^\infty e^{-\pi x^2} \cos(2\pi xw)dx, w\in\mathbb{R} $$ I encountered this ...
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1answer
29 views

Show a given analytic function is constant

Suppose that $f$ is analytic on some region $R\in\mathbb{C}$. If Im$(f)$ = $k\cdot$Re$(f)$ for some nonzero constant $k\in\mathbb{C}$, then show that $f$ is constant on $R$. I know that if $f'(z)=0$ ...
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34 views

Why are these integrals the same?

For $\xi, h \in \mathbb{R}^n$, using substitution $h = \rho\omega$, $\rho = |h|$, and $\omega = \displaystyle \frac{h}{|h|}$, we have $$\int_{\mathbb{R}^n} \frac{|e^{i2\pi\xi\cdot ...
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33 views

Laplace transforms of powers of cosine

During the past several hours, while studying the Laplace transform, I've started wondering what \begin{equation} \mathcal{L} \{ \cos^n(at)\}(s) \end{equation} would look like – since it won't ...
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60 views

Evaluate for $t\in \mathbb{R}$ $\int_{-\infty}^\infty{e^{itx} \over (1+x^2)^2}dx$

Evaluate for $t\in \mathbb{R}$ $$\int_{-\infty}^\infty{e^{itx} \over (1+x^2)^2}dx.$$ Here is what I have done: Let $f(z)={e^{itz}\over (1+z^2)^2}$. This has two poles $z=i$ $z=-i$ and an essential ...
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22 views

>$f(z)$ has a zero of order $n$ at $z_0 \iff F(z)=f'(z)/f(z)$ has a simple pole at $z_0$. Find the residue of $F$ at $z_0$.

$f(z)$ has a zero of order $n$ at $z_0 \iff F(z)=f'(z)/f(z)$ has a simple pole at $z_0$. Find the residue of $F$ at $z_0$. This Questions suggests that the residue is n However, could somebody ...
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75 views

Show that the series defines and entire function, $\sum_{n=1}^\infty {{z(z+1)\cdots (z+n-1)}\over n^n}.$

Show that the series defines and entire function, $$\sum_{n=1}^\infty {{z(z+1)\cdots (z+n-1)}\over n^n}.$$ I that that an entire function is one that is analytic at every point in the plane. This ...
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25 views

Find a conformal mapping from the quarter-disk $Q=\{ |z|<1 : rez>0,im z>0 \}$ onto the upper half plane set $U=\{im z>0\}$

Find a conformal mapping from the quarter-disk $Q=\{ |z|<1 : rez>0,im z>0 \}$ onto the upper half plane set $U=\{im z>0\}$ I'm guided through this problem: First I need to find the ...
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3answers
20 views

Calculating the limit of complex norm squared

Let $g(z)= z \overline{z}$. Prove that $g^\prime(z_0)$ exists $\iff z_0 = 0$. I already proved $(\Leftarrow)$, how do I do the forward direction $(\Rightarrow)$? I tried using the definition of the ...
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33 views

Proving an analytic function is rational [duplicate]

If f is analytic in $|z| \leq1 $ and satisfies $ |f(z)| = 1$ on $|z| = 1$ prove that f is a rational function. I need a clue how to think about this? should i use the fact that f can be written as a ...
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22 views

Pole on the Real Axis of Complex Integral

Save me! SOS!. Please see image. I'm completely baffled about how to go about solving this. Please can you explain the how you get -pi*i from the 4th line? Can you please explain by this paragraph as ...
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1answer
33 views

Show that: $\int_\Gamma \frac{\log(z+i)}{1+z^2}dz\rightarrow 0 $ along the semicircle $\Gamma$ when we take $R\rightarrow \infty$

This is part of a bigger problem. My problem tells me I can use this fact. But I want to prove it. Show that: $$\int_\Gamma \frac{\log(z+i)}{1+z^2}dz\rightarrow 0 $$ along the semicircle $\Gamma$ ...
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32 views

Residue of $f(z)=e^{-1/z^2}$ using it's pole order

Find the residue of $f(z)=e^{-1/z^2}$ at $z=0$ Can somebody check both methods? I wrote out the series of $$e^z=1+z+\frac{z^2}{2}!+...$$ and subbed in $-1/z^2$ instead of $z$ to get: ...
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57 views

How many roots does the polynomial $p(z) = z^8 + 3z^7 + 6z^2 + 1$ have inside the annulus $1 < |z| < 2$?

How many roots does the polynomial $p(z) = z^8 + 3z^7 + 6z^2 + 1$ have inside the annulus $1 < |z| < 2$? I know I can use Rouche's Theorem. I'm just not sure how. It states that $|f(z) − g(z)| ...
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22 views

On the zeroes of sine complex function, and a search for a special sequence, following Riemann's approach

If there are no mistakes from the Fourier expansion series for the fractional part function we can write, using a substituion, that for $1<x<e^2$ with uniform convergence $$\frac{1}{2}\log ...
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25 views

$L^2$ convergence of Taylor series of a holomorphic function

I am reading Otto Forster's book "Lecture on Riemann surfaces" and on pages 109-110, he introduces the space $L^2(D,\mathcal{O})$ of holomorphic square-integrable functions $f:D\to \mathbb{C}$ (where ...
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1answer
38 views

Integral similar to fresnel integrals

$$\int_{0}^{+\infty} \frac{e^{-r^2}}{r^2-i\gamma^2} dr = ?$$ I tried the normal semicircular contour integrals, but there is always a problem with the exponential when I close the contour. This post ...
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1answer
46 views

Existence of analytic function with $f(3/4)=3/4$ and $f'(2/3)=3/4$ [on hold]

Let $D=\{z\in \mathbb{C}:|z|<1\}$. Then there exists an analytic function $f:D \to D$ with $f(3/4)=3/4$ and $f'(2/3)=3/4$. Any help would be appreciated.
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148 views

Correcting Gauss's proof of FTA. Need verification

In his doctoral thesis, Gauss gave a proof of fundamental theorem of algebra for real polynomials, based on geometric arguents. Later in his life he expanded the proof to complex polynomials. A nice ...
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1answer
51 views

Taylor expansion of $\frac1{\|x-y\|}$

Let $0\neq y\in \mathbb{R^3}$ define a function $f$ on $\mathbb{R^3}$ as $$ f(x) = \frac1{\| x-y\|} $$ What are derivatives of $f$ in zero? Or equivalently, what is the Taylor series of $f$ at ...
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13 views

proof that ${f^k}$ if normal family. Julia Set

Let $f^k=f \circ f \ldots\circ f$ and $T$ Mobius transformation. I like proof that: $\{f^k\}_k$ is a normal family $\Leftrightarrow $ $\{T\circ f^k \circ T^{-1}\}_k$ is a normal family. I dont know ...
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44 views

What is $\frac{\partial^2}{\partial \bar{z}\,\partial z}\log|z|^2$?

Consider the function $$\Bbb C-\{0\}\to\Bbb R,\quad z\mapsto\log|z|^2.$$ What is $$\frac{\partial^2}{\partial \bar{z}\,\partial z}\log|z|^2?$$ Try: I am no sure if the second step is justified, but ...
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1answer
58 views

Intuition for integrating $1/z$ around the unit circle

Most standard results in complex analysis depend on the fact that $\int_C 1/z \ d\gamma = 2\pi i.$ Evaluating the integral by definitions is not hard, but also not very illuminating. The result ...
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1answer
12 views

Big-O Notation?

The problem is to to evaluate the following contour integral along a path $C$ defined/parameterized as $z(t)=εe^{it}$: $ \int_C \frac{e^{iz}}{z} dz$ The solution for the problem proceeds to say ...
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1answer
28 views

Showing that complex functions with the same derivative on the unit disc differ by a constant

I have from class: If $U\subset \mathbb{C}$, convex and $f:U\rightarrow \mathbb{C}$ is holomorphic, then f has a primitive. My proof is: The fact that $D(0,1)$ is clearly convex and ...
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94 views

Difficulty evaluating complex integral

The integral along the path $\gamma(t)=e^{2ti},\;t\in[0,2\pi]$ is $\begin{equation*} \int_{\gamma}\frac{1}{z^{2}-1}dz \end{equation*}$. I approached this like a real integral in the hopes things ...
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26 views

Evaluation of complex integral?

I'd like to verify the result of this integral, or find if I've made a mistake. In the following, $\mathbf x, \mathbf a, \mathbf b$ are all real vectors in $\mathrm R^3$. I do the following: group ...
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26 views

Do we care about singularities on the path when using the Residue theorem? (And other theorems in complex analysis)

Take for example the Residue theorem, if we want to calculate a line integral around the unit circle say and the function isn't defined at some points on the unit circle say $f(z)=1/(z-1)$ then if we ...
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1answer
25 views

How to find and classify the singularities of $\frac{e^{-z} \sin(2(z-1)^2)}{(z^2-4)(z-1)^2}$?

How to find and classify the singularities of $$\frac{e^{-z} \sin(2(z-1)^2)}{(z^2-4)(z-1)^2}$$ Here is what I have: I think the singularities are all isolated and are located at $z=1,\pm2$. I have ...
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1answer
20 views

How to tell the minimum point of a higher order function?

I was following an example in my Complex Analysis textbook where we are finding the number of zeroes of the function $f(z) = z^3-2z^2+4$ in the first quadrant. They begin by saying on the real axis ...
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16 views

How to find $\int_{\Gamma} \cos z \sin z dz$ where $\Gamma = \pi t + (1-t)i$ for $t \in [0,1]$?

How to find $\int_{\Gamma} \cos z \sin z dz$ where $\Gamma = \pi t + (1-t)i$ for $t \in [0,1]$? I subbed in to get $$(\pi-i)\int_0^{1} \cos (\pi t + (1-t)i) \sin (\pi t + (1-t)i) dt$$ which I found ...
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1answer
35 views

Modulus of roots of polynomial tend to infinity

Define $f_n:\mathbb{C}\to\mathbb{C}$ and $(\alpha_n)$ such that:$$f_n(z)=\sum_{k=0}^n \frac{z^k}{k!}$$ and $f_n(\alpha_n)=0$. Prove $|\alpha_n|\to\infty$ as $n\to\infty$. I guess this makes sense ...
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How to find the largest $R$ such that the Laurent series of $f(z)=\frac{2}{(z^-1)}+\frac{3}{2z-i}$ about $z=1$ converges for $0<|z-1|<R$?

How to find the largest $R$ such that the Laurent series of $$f(z)=\frac{2}{(z^2-1)}+\frac{3}{2z-i}$$ about $z=1$ converges for $0<|z-1|<R$? What I have done so far: ...
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1answer
35 views

Continuous function - $|\int_C f(z)dz| \leq 4$

Question : Let the continuous function $f : \mathbb{C} \to \mathbb{R}$ such that $|f(z)| \leq 1$ and $C$ be the unit circle in the positive direction. Show that $\left|\int_C f(z)dz\,\right| \leq 4$. ...
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45 views

Estimate trigonometric functions with complex argument

I would like to prove the following estimates $\vert \sin(z)\vert\leq \sinh(s)$ and $\vert \cos(z)\vert\leq \cosh ( s )$ ,where $z\in D_s(0)\subset\mathbb{C}$ and $D_s(0)$ denotes the disc with ...
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29 views

Conformal maps and elliptic functions

Can someone please help me with the following complex (?) analysis question? I'm given a triangle $T$ (say with angles $30^{\circ}$, $60^{\circ}$ and $90^{\circ}$) and a conformal map from $T$ onto ...
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19 views

Bounding a complex function given a condition on its input

Let $z\in\mathbb{C}$ such that $\vert z\vert = R$. Then I want to bound the function $$f(z) = (z^2+4)^2(z^2+9)$$ from below. So I have \begin{align} \left\vert (z^2+4)^2(z^2+9)\right\vert ...
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23 views

Rearrangement of Complex Sin and Cos

From my complex numbers course notes, there is the following derivation: The definitions of sin and cos I'm very comfortable with, but I cannot see how we get from the definition to the given ...
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1answer
23 views

Estimation for points in a neighbourhood of a root of a polynomial

Let $p(x)$ be a polynomial with complex coefficients and $p(\tilde x)=0$. Choose $\delta>0$ small enough, such that $\tilde x$ is the only root of $p$ in $B_\delta(\tilde x)$. I want to show that ...
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1answer
27 views

Expansion of $f = \sum_{n=1}^\infty \frac{1}{2^n} \frac{z^n}{1 - z^n}$ in power series around $z = 0$

Let $f = \sum_{n=1}^\infty \frac{1}{2^n} \frac{z^n}{1 - z^n}$, for $z \in \mathbb C \setminus ${$z \in \mathbb C: \exists n \geq 1,\quad z^n = 1$}. By the ratio test, the series converges when ...
2
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1answer
42 views

Singular Points and their Classification

Suppose I have these functions: $\frac{1}{z-z^3}$ $\frac{1}{(z^2+4)^2}$ $\frac{e^z}{1+z^2}$ I need to find the singular points of these functions, and classify them ($\infty$ included). So far, I ...
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2answers
53 views

Jordan Lemma - Complex Integral

How does the all of the LHS equal zero? $$\lim_{R\to\infty} \int_{H_R} \frac{e^{imz}\,dz}{a^2 + z^2}=0$$ Please see this image for more full question and context.
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0answers
32 views

What is an exponential singularity?

In Ken Ono's Lecture notes on Harmonic Maass forms and Mock Modular Functions (here) the author uses the term "exponential singularity". What does this mean? Thanks a lot!
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19 views

Theorem on meromorphisms - complex analysis

S is a sequence converging to $z_0$ and f is analytic is a disk centred at $z_0$ show either that f extends meromorphically on some neighbourhood of f(z) or else for any $L \in \mathbb{C}$ there ...
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1answer
74 views

Proving $\int_{-\infty}^{+\infty} {{\cos(mx)}\over(x^2+a^2)(x^2+b^2)}dx={\pi(ae^{-mb}-be^{-ma})\over ab(a^2-b^2)}$

Show that $$\int_{-\infty}^\infty {{\cos(mx)}\over(x^2+a^2)(x^2+b^2)}dx={\pi(ae^{-mb}-be^{-ma})\over ab(a^2-b^2)}$$ where $a,b,m>0$ and $a$ is not equal to $b$. I already know that ...
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0answers
24 views

Proof that $0$ is not in the codomain of the following complex functions

I have the functions $f:[0,1]\times\mathbb{C}\backslash\{0\}\rightarrow\mathbb{C}$, $f(x,y)=(1-x)y+xy/|y|$, $g:[0,1]\times\mathbb{C}\backslash\{0\}\rightarrow\mathbb{C}$, ...