# Tagged Questions

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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### Question on radius of convergence

Can anyone help me with the following problem: I have a solid geometric picture of what is going on in my head, but I can't seem to turn that into a proof.
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### not following two steps in proof that $\int_{0}^{\infty}\cos(x^2) = \frac{\sqrt{2 \pi}}{4}$

Hi: I'm reading some notes I found on complex analysis on the internet. In the example, they prove that $$\int_{0}^{\infty} \cos(x^2) = \int_{0}^{\infty} \sin(x^2) = \frac{\sqrt{2\pi}}{4}.$$ I ...
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### If $f$ is the limit of polynomials with only real zeros, then all zeros of $f$ are real

Problem Let $f$ be a non-constant entire function. Suppose that there is a sequence of polynomials ${P_n(z)}$, $n=1,2,...$ such that $P_n(z)$ converges uniformly to $f$ on every bounded set ...
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### Do asymptotically equivalent coefficients survive convolution at least in Θ?

This is a follow-up question to this one where I asked if asymptotic equivalence of coefficients carried over after convolution, resp. why this was not the case. Answerer Daniel Fischer proposed that ...
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### holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
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### Cauchy-Riemann and Analytic Functions

Using the Cauchy-Riemann conditions, tell if $f(z) = z^{*}$ is analytic I have tried this: $Z = x + iy$ $f(x + iy) = Z^{*} = x - iy$ $U(x,y) = x$ $V(x,y) = -y$ $U_x = 1$ Deriving respect to $x$ ...
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### Contour Integration $\int_0^1\frac1{\sqrt[n]{1-x^n}}dx$

I want to compute: $$\int^{1}_{0}\frac{1}{\sqrt[n]{1-x^n}}dx$$ for natural $n>1$ using Residue Calculus. I am thinking of using some kind of a keyhole or bone contour that could go around the $n$...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 * ...
### Integrating $\int_0^1 dx\,\ln(x-a)/(x-b)$ paying attention to cuts.
I am trying to compute the following integral, for complex $a$ and $b$ $$\int^1_0 dx \frac{\ln(x-a)}{x-b}$$ by turning it into something in terms of dilogarithms. But for certain values of $a$ and \$...