# Questions tagged [complex-analysis]

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, isolated zeros principle...

34,667 questions
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### Why do $x^2+y^2=\left(3+2i\right)^2$ and $x^2+y^2=25$ give the same circle?

Maybe I just been up for too long but I just cant get why: $$x^2+y^2=\left(3+2i\right)^2$$ and $$x^2+y^2=25$$ give the exact same circle, why aren't the graphs different? Shouldn't the second one ...
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### Why $\Im \frac{1}{1+e^{-s + i a }}=\frac{\sin(a)}{\cos(a)+\cosh(s)}$?

Why this $$\Im \frac{-2}{1+e^{-s + i a }}$$ equals to this expression: $$\\\ \frac{\sin(a)}{\cos(a)+\cosh(s)}$$ I was trying to evaluate the Fourier transform of a hyperbolic function and my ...
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### How can I minimize the real part of the roots of this function involving both $x$ and $e^x$ terms?

The question I have a function $D(s) = s^2 + c s + k + K_d s e^{-s} + K_p e^{-s}.$ The values of $c$ and $k$ are fixed, but I can choose $K_d$ and $K_p$. How do I choose these two values in order to ...
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### Given two entire functions such that $\operatorname{Re}(f(z))\le a·\operatorname{Re}(g(z))$, prove that $f(z)=a·g(z)+c$

Let $a\in \mathbb{R}$ and $f, g$ two entire functions such that $\operatorname{Re}(f(z))\le a·\operatorname{Re}(g(z))$. Prove that there exists a constant $c\in \mathbb{C}$ such that $f(z)=a·g(z)+c$ ...
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### Proving $f_n(z)=\frac{nz}{1+n^3z^2}$ converges uniformly

Show that the sequence of functions $f_n(z)=\frac{nz}{1+n^3z^2}$ converges uniformly on the set $E=[1,\infty]$. $\lim_{n\to\infty}\frac{nz}{1+n^3z^2}=0$ so it converges pointwise to 0. So I am going ...
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### Euler's definition of complex analysis

Can someone help? Need to find the answer of $$\frac{e^{iz} + e^{iz}}{2} \ .$$
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### Singular points in Complex analysis

Can someone help? Need to find the singular points of $$\frac{z+1}{(z^3)(z^2+1)}$$ where $z$ is a complex variable
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### Laurent series and tensor

Let us begin with the complex vector space \begin{equation} V_{z}=\Big\{\omega\in \mathbb{C}[[z,z^{-1}]]dz\ \mid \operatorname{Res}_{z=0} \omega (z)\Big\} \end{equation} We could define the tensor ...
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### $f : \Bbb{C} \to \Bbb{C}$ is an entire function s.t. $|f(z)| \to \infty$ as $|z| \to \infty$. Prove $f$ is a polynomial.

Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be an entire function such that $|f(z)| \rightarrow \infty$ as $|z| \rightarrow \infty$. Prove that $f$ is a polynomial by following the steps below. (a) ...
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### Proving $\{f_n(z)=e^{-(z-n)^2}\}$ is uniformly convergent [on hold]

Show that the sequence of functions $\{f_n(z)=e^{-(z-n)^2}\}$ converges uniformly on the set $E=D(0,1)$. $|e^{-(z-n)^2}-0|=|e^{-z^2}.e^{-n^2}.e^{2nz}|\leqslant |e^{-n^2}.e^{2nz}|$ However I do not ...
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### Why is $res(\Gamma(s)x^{-s}\zeta(2s)|s=\frac{1}{2}) = \frac{\Gamma(\frac{1}{2})}{2x^{\frac{1}{2}}}$?
Why is $res(\Gamma(s)x^{-s}\zeta(2s)|s=\frac{1}{2}) = \frac{\Gamma(\frac{1}{2})}{2x^{\frac{1}{2}}}$? where res means the residue of the function? I know $\zeta(s)$ has a pole at $s=1$ but i can't ...