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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Computing the $n^{th}$ coefficient of the power series representing a given rational function

Is there a easy way to compute the coefficients of the power series which represents \begin{equation*} \frac{x - x^k + x^{k+1}}{1-2x + x^k - x^{k+1}}. \end{equation*} I am currently solving this ...
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$\int_{-\infty}^\infty\frac{1}{(x^2+a^2)^3}dx=\frac{3\pi}{8a^5}$ for $a>0$

I've been trying to show that $\int_{-\infty}^\infty\frac{1}{(x^2+a^2)^3}dx=\frac{3\pi}{8a^5}$ for $a>0$ using complex analysis methods. But for some reason I can't get it to come out. Perhaps ...
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Using Cauchy Integral Formula $\int_C \frac2{z^2 -1}dz$

I want to understand why I can't use Cauchy Integral Formula for the following problem: $$\int_C \frac2{z^2 -1}dz\text{ on the contour } |z-1|=\frac12$$ Now it says that I need $f$ to be analytic ...
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Show that $f(z)=z+a_2 z^2$ is univalent in $\mathbb{D}=\{z∈\mathbb{C}:|z|<1\}$ if and only if $|a_2 | \leq 1/2.$

Show that f(z)=z+a_2 z^2 is univalent in D={z∈C:|z|<1} if and only if |a_2 |≤1/2. My solution: (If part): Suppose f(z)=z+a_2 z^2 is univalent in D. By definition, we know that f(z_1 )=f(z_2 ) ...
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If $\Re(f)$ is bounded then f is constant.

I have to solve following problem If $\Re (f)$ is bounded above or below for a function $f$ holomorphic on $\mathbb{C}$ then $f$ is constant. My attempt: If there is $M$ such that $\Re(f) \le M$,...
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Is the Gamma Function multivalued??

Consider the definition of the Gamma function $$\Gamma(s) = \int_{0}^{\infty}\left[x^{s-1}e^{-x} \right] dx$$ Clearly: $x^{s-1}$ may have multiple defined values for $s$ if $s-1$ is rational or ...
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Is the difference in the hypotheses of these two statements relevant?

The following is a problem from Conway's Functions of One Complex Variable, and my proof: Let $G$ be a region and suppose that $f:G \rightarrow \mathbb{C}$ is analytic. Show that if $f(z)$ is real ...
Let $\Phi(z)$ be an entire function of finite exponential type. The indicator function of $\Phi(z)$ is defined as  h_{\Phi}(\theta)=\overline{\lim_{r\rightarrow\infty}}\frac{\ln|\Phi(re^{i\theta})|}...
$f:D \rightarrow \mathbb C$ holomorphic, $D$ a convex set, $Re(f'(z))>0$. Prove that $f$ is injective [duplicate]
$f:D \rightarrow \mathbb C$ holomorphic, $D$ a convex set, $Re(f'(z))>0$. Prove that $f$ is injective. What I tried: Assume that $f$ is not injective. So there are $a,b \in D$ so that $f(a)=f(b)$. ...