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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Let $f: \mathbb{D}\to \mathbb{D}$ be analytic. Prove that $|f'(z)|\leq \frac{1}{1-|z|}$. [duplicate]

I am stuck on a qualifying exam problem while I am studying. I was thinking I should use the Schwarz-Pick Lemma, but I can't seem to get the result. Let $f: \mathbb{D}\to \mathbb{D}$ be analytic, ...
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2answers
29 views

Determine the nature of singularities and calculate the residue of $f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3}$

$$f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3},\;\;\;\;\;\;\; \mathrm{Res}[f(z),0]$$ I am having trouble determining the nature of singularities. This is what I managed to do: ...
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22 views

Please justify this statement: “[A] holomorphic function is (n+1)-to-1 near a zero of its derivative of order n”.

Another member of the community posted this in one of their answers to a question a few years back and I can't seem to understand why this is true. Help?
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2answers
26 views

Usage of the term $\arg(z)$

Consider the complex number $z = -1 - i$. Is it mathematically correct to say that $\arg(z) = 5\pi/4$? Sure, $5\pi/4$ is not the principle argument of $z$, but it is an element of the set $\arg(z)$. ...
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9 views

Connected components of unbounded portion of entire function.

Suppose $f$ is an entire complex analytic function and $R$ a positive real number. Define the set $E:= \{z\in\mathbb{C};|f(z)| < R\}$ to be the set of $z$ whose image is bounded by $R$. I want to ...
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1answer
71 views

Why does convolution not maintain asymptotic equality of coefficients?

Assume I have four (generating) functions $f$, $f'$, $g$ and $g'$. If that is interesting, we can assume that they all share the same radius of convergence $\rho > 0$. In addition, we know that ...
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1answer
29 views

Quotient of 2 holomorphic functions which may be holomorphic

Let f and g be two holomorphic functions on a domain $\Omega$. Suppose that $\frac{f}{g}$ is always finite (while g can be zero at some points). Is it true that then $\frac{f}{g}$ is holomorphic? ...
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37 views

How do you calculate the Riemann zeta function of a complex number given the complex contour integral?

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
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22 views

Find the Laurent Expansion of $f(z)=\frac{1}{z+i}$

Find the Laurent Expansion of $f(z)=\frac{1}{z+i}; f(z)=\frac{1}{(z-i)^2}$ and $f(z)=e^{(z-1)^-1}$ Good evening, I have been trying to solve the above exercises. However, I'm not sure if my procedure ...
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22 views

Conformal mapping between regions symmetric across the real line

In Conway's Functions of One Complex Variable, the section on the Riemann Mapping Theorem has the following exercise: Let $G$ be a simply connected region which is not the whole plane, and suppose ...
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1answer
23 views

Sketching regions is complex plane

When sektching the region $\left|\frac{2z-1}{z+i}\right|$$\geq$1 on the argrand diagram, how should we go about identifying the region, should we take $\left|2z-1\right|\geq\left|z+i\right|$ or ...
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1answer
18 views

Describe the family of analytic functions with the following properties:

Find the family of all functions $f$ analytic in $\mathbb{D}$ (the open unit disk) and continuous on $\overline{\mathbb{D}}$ such that $|f(z)|=e^{\text{Re}(z)}$ for all $z\in\mathbb{D}$. My intuition ...
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0answers
33 views

Entire function with $L^2$ modulus is identically zero [duplicate]

I want to show that if $f$ is entire and $\int_{\mathbb{R}^2}\left| \:f\: \right|^2 < \infty$, then $f \equiv 0$. I was thinking of assuming $f$ is not identically zero; then, since a bounded ...
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47 views

What does this complex contour integral represent?

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
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1answer
26 views

Convergence of an infinite power

There are complex numbers $z$ and $w$ for which $$\lim_{n\rightarrow\infty}z\uparrow\uparrow n=w$$ where $\uparrow\uparrow$ is the tetration symbol, e.g. $z=\sqrt{2}$ and $w=2$. Are there complex ...
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0answers
35 views

Contour integration with merged pole/branch-cut type behavior?

I have the expression $$f(z)=\frac{-i}{\sqrt{z^2-a^2}},$$ where $a$ is a purely real number and $z$ is a complex variable. Numerical plotting gives the following. This leads me to the following ...
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1answer
25 views

winding number in several complex variables

Is there any analogue of the concept of winding numbers in the theory of several complex variables? If so, can anyone provide me references for studying it?
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3answers
75 views

Solving the equation $(z-2)^{4}+(z+1)^{4}=0$

$(z-2)^{4}+(z+1)^{4}=0$ I tried starting by solving $z^{4}=1$ with the solutions being , $1cis (\frac{n\pi }{2})$, where $n = -1, 0, 1, 2$ I am unsure about how to proceed from here, I tried to ...
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0answers
45 views

What does this complex contour integral mean? [on hold]

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
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1answer
18 views

Extrema of the set of values $|f(-1/2)|$ for analytic functions $f \colon \mathbb{D} \to \mathbb{D}$

I have a past qual question here: consider the set $S = \{ |f(-1/2)| \colon \textrm{$f \colon \mathbb{D} \to \mathbb{D}$ is analytic and has a triple zero at the origin} \}$, where here $\mathbb{D}$ ...
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1answer
49 views

given analytic $f(z)$ in $f(z)/(1-z)$ , derivative $f '(z)$ seems to have singularity at $z=1$

Quick version: I want $f'(1)$, where $$F(z)=\frac{f(z)}{1-z}$$ with $f$ analytic at $z=1$. But when I follow a seemingly valid line of reasoning, I reach the conclusion that $f'(z)$ is not analytic ...
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1answer
33 views

Assume that $f$ is analytic and one-to-one on $\mathbb{D} = \{z : |z| < 1\}$ and $f(z) = z + z^2g(z),$ where $g$ is analytic in $\mathbb{D}.$

Assume that $f$ is analytic and one-to-one on $\mathbb{D} = \{z : |z| < 1\}$ and $f(z) = z + z^2g(z),$ where $g$ is analytic in $\mathbb{D}.$ Prove that if $f(\mathbb{D})⊂\mathbb{D}$ or ...
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1answer
69 views

Evaluation of Sum of $ \sum_{n=1}^{\infty}\frac{\sin (n)}{n}$.

If $\displaystyle S = \sum_{n=1}^{\infty}\frac{\sin (n)}{n}.$ Then value of $2S+1 = $ Using Fourier Series Transformation I am Getting $2S+1=\pi$ But I want to solve it Using Euler Method and Then ...
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1answer
53 views

Calculating $\int_0^\pi \sin^2t\;dt$ using the residue theorem

I want to use the residue theorem to calculate $$I:=\int_0^\pi \sin^2t\;dt$$ Since $\sin^2$ is an even function, we've got $$I=\frac{1}{2}\int_0^{2\pi}\sin^2t\;dt$$ The solution of this exercise ...
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4answers
75 views

Help with Complex integration

I have to calculate the following integral $$\int_{-\infty}^{\infty} \frac{\cos(x)}{e^x+e^{-x}} dx$$ Anyone can give me an idea about what complex function or what path I should choose to calculate ...
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1answer
64 views

Help on a tough summation from Rudin?

I'm having a tough time deriving (4) from the bracketed expression in (3) shown in the photo. I've been futzing with partial sums of geometric series and binomial expansions for a while now with no ...
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64 views

Strategy verifying Riemann Hypothesis? [on hold]

The basic strategy for verifying the Riemann Hypothesis Count all of the zeros of $\zeta(t)$ for $0 < t < T$ Compute an upper bound on the number of zeros of the zeta function which lie in the ...
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95 views

General method to “naturally interpolate” to a complex map?

Given a region of the complex plane and a map $z \to f(z)$, is there a general way to "naturally interpolate" the point $z$ to $f(z)$ in such a way that the movement follows a "natural" smooth path ...
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0answers
25 views

Uniformly analytic functions

Consider the following definition: Let $\Omega$ be an open set of $\mathbb{R}_x^n$, $x = (x_1, ..., x_n)$. A $\mathcal{C}^{\infty}$-function $\varphi(x)$ on $\Omega$ is said to be uniformly analytic ...
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1answer
34 views

Mean value theorem for harmonic

In Problems and Solution in Mathematics by Ta-Tsien, exercise 5123, the mean value theorem is used as: \begin{equation} \text{log} |F(0)| = \frac{1}{2 \pi} \int_0^{2\pi} \text{log}|F(re^{i\theta})| ...
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1answer
18 views

not following a step in ash and novinger example of analytic but does not have primitive

I'm trying to self-study complex analysis and am currently reading the book "complex analysis" book by ash and novick. 0n the top of page 14, they write that , if $f(z) = \frac{1}{z}$ and the path ...
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0answers
37 views

residue theorem in complex analysis(Rudin)

In P.224, Rudin's real and complex analsysis, I doubt an equation (3). The full statement containing equation (3) is following : If $\Gamma$ is a cycle and a $\notin$ $\Gamma$*, then ...
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1answer
26 views

Showing that derivative of conjugate is conjugate of derivative, using chain rule

I'm trying to show that the derivative of the conjugate is the conjugate of the derivative, i.e. $\dfrac{d[f(x)^*]}{dx} = [\dfrac{df(x)}{dx}]^*$, using the chain rule. Calling the conjugate * ...
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1answer
13 views

Can a Loxodromic Transformation be the Composition of 2 Reflections?

I've been reading "Visual complex Analysis," and it proved that a Mobius transformation whose multiplier is real can be written as the composition of two reflections in circles/lines (i.e., ...
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2answers
63 views

Complex Power Series

So, I'm trying to find the power series of ${1\over 1-z+z^2} around the point z=0.$ After some rather easy algebra I've determined the expression to be $${1\over z-(1+i\sqrt{3})/2} {1\over ...
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2answers
36 views

Show elementarily that $\lim_{R\to\infty}\int_{\Gamma_1} \frac{e^{iz}}{z} = 0$

Context: I am trying to show that $\int_0^\infty x^{-1}\sin x dx = \frac{\pi}{2}$ using complex analysis, by first integrating $\oint_{\Gamma} z^{-1}e^{iz}$, where $\Gamma$ is a closed contour ...
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16 views

Using residue theorem on a double pole in which the numerator is a function of the variable squared.

If I have an integral which is evaluated wrt $\mu$ that is evaluated using the residue theorem and there is a double pole at the origin, I want to differentiate the function wrt $\mu$ to find the ...
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1answer
25 views

Constructing an analytic continuation

I'm hoping someone could verify my answer to the following problem: Consider a function $f$ that is continuous for $Im(z) \geq 0$ and analytic for $Im(z) > 0$. Furthermore, assume that $f$ ...
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0answers
44 views

Critique of complex analysis proof

I'm working on the following complex analysis problem and am wondering if someone could critique my proof: Suppose that $f$ is an analytic function on some domain $D$ and that there exists a ...
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2answers
32 views

Does a bounded continuous function map Cauchy sequences to Cauchy sequences?

I only ever see the example of $f:(0,1]\rightarrow \mathbb{R}$ where $f(x)=\frac{1}{x}$as that of a continuous function that does not map Cauchy sequences to Cauchy sequences. Are there examples of ...
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3answers
67 views

poles of a polynomial

What are the poles of a polynomial? Are they the same as the roots?
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73 views

Computing integral using complex analysis methods

I'm trying to compute the integral $$ \int_0^{\infty} \frac{\ln(x)}{x^2 + 1} \, dx $$ using complex analysis methods. We haven't learned residue calculus yet though, only contour integrals up ...
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1answer
59 views

Computing a very messy contour integral

I'm hoping that someone might be able to help me with the following problem. I'll walk through my current work and indicate where I'm stuck. Compute the contour integral: $$ \oint_{|z-1-i| = ...
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0answers
25 views

Domain of convergence Taylor series [closed]

What would be the domain of convergence of the taylor series of : 1) $\frac{sinz}{z^2+3}$ at $z=0$ 2) $\frac{z+5}{(z-1)(z-4)}$ at $z=2$ 3) $zcoth6z$ at $z=0$ How do I do this? Thanks
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1answer
56 views

Contour integral method

Let $f(z)=z^5-3iz^2+2z-1+i$. Evaluate the integral of $\frac{f'(z)}{f(z)}$ around a contour $C$ where $C$ encloses all the zeroes of $f$. I'm not sure what to do here. It seems unlikely I should be ...
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3answers
44 views

meromorphic function with a pole on the unit circle diverges

Let $f$ be a meromorphic function in a neighborhood of the closed unit disk $\bar{\mathbb{D}}$. Suppose that $f$ is holomorphic in $\mathbb{D}$ and $$ f(z) = \sum_{n=0}^\infty a_n z^n $$ for $z \in ...
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1answer
44 views

Find the set of $z$ which satisfies the given equation

Let $w \to w^{a}$ be the principal branch of the power function defined for $|\mathrm{Arg}(w)| <\pi$. Find the set of all values of $z\in \mathbb{C}$ such that the following identity holds for ALL ...
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29 views

Question about Maximum Modulus Principle [duplicate]

Let $f_1, f_2, \ldots ,f_n$ be holomorphic functions on a region $\Omega$. Show that if $\phi (z)=|f_1(z)|+|f_2(z)|+\ldots +|f_n(z)|$ attains a maximum value on $\Omega$, then $f_i$ is constant for ...
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1answer
45 views

Complex Fourier Series and using the square norm

Find the complex Fourier series of $f(x)=e^{(-πx/2)}$ on $-π < x < π$ Discuss the significance of $|C_n|$ in the solution. I've tried so far Using the Complex Fourier Series: $$ %% ...
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0answers
29 views

Showing that $\log|f|$ is harmonic given that $f$ is analytic [closed]

Suppose that $f$ is an analytic function. How would I show that $\log|f|$ is harmonic?