# 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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### Showing a complex function is nowhere differentiable in a certain disc

I have a function and I am asked to prove that it is nowhere differentiable on an open disc. I found the cauchy riemann equations and saw that is is satisfied at the origin. I don't know what to do ...
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### Help With Bromwich Inversion Formula Proof

To prove(copied from handwritten notes so possibly wrong): Bromwich Inversion Formula. Fix $x_0∈ℝ$. If $F$ is complex analytic on $\{z:\Re z > x_0\}$ and for every $x>x_0$, $y↦ F(x + iy )$ ...
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### Under what conditions do you use that $\operatorname{Res}{(f(z)/g(z))}=f(z_0)/g'(z_0)$?

In complex analysis, this seems to be a really helpful way to avoid having to expand out Laurent series. I am unclear, however, when it is appropriate to use this property. In specific, I'm worried I ...
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### Residues of $z^2\sin(\frac{1}{z})$

I must find the residues of $z^2\sin(\frac{1}{z})$ at $z = 0$. Since $z = 0$ seems to be an Essential Singularity, i'm not sure how I can continue to find the residue of the function. Usually I am ...
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### Harmonic functions proof

I don't understand here why: $2(\Delta(u_x)^2+\Delta(u_y)^2) \geq 0$. Here $\Delta= \nabla^2, \quad u'_x=u_x$ etc
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### A problem about elliptic functions

I am trying to solve some problems in complex analysis, but I am not succeeding in the following problem. Suppose that $f$ is a function with the following properties: $f$ is non-constant; $f$ is ...
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### A Funtional equation in Complex variables

I have been stuck on this problem for a long time : If $f(z)=u(x,y)+iv(x,y)$ , prove that a. $f(z)=2u(z/2,(-iz)/2) +$ constant b.$f(z)=2iv(z/2,(-iz)/2) +$ constant This result seems very ...
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### $f=u+iv$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$?

$f(z)=u(x,y)+iv(x,y)$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$? I tried to interprete $xu+yv$ as some part of a new function, for example, as the real part of $\overline{z}f$，but this ...
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### Complex function, analyticity domain

Find the function domain of analyticity i)$f(z)=\frac{z^2}{z-3}$ ii)$f(z)=ze^{-z}$ To check the domain of analyticity of a function, I only need to replace $z=x+iy$ and check the conditions of ...
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### Find an analytic function $f:\mathbb{C}\setminus\{-1\}\rightarrow \mathbb{C}$ such that $f'(z)=\frac{z}{z+1}$ or show that no such function exists.

I have a guess that the function does not exist. But I dont know how to show it. I have been suggested to look at the following theorems: 1): If f is entire, then f is everywhere the derivative of an ...
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### Is there a measure invariant with respect to the Möbius transformation?

I would like to use a measure ${\rm d} \mu (z)$ on ${\mathbb C}$ so that for any $f(z)$ $$\int_{\mathbb C} f(z) {\rm d} \mu (z)$$ is invariant under Möbius transformations. Taking the ...
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