# 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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### How to find the singularities of the function $z(1-e^{\frac{1}{z}})$ and classify them

Find the singularities of the function $z(1-e^{\frac{1}{z}})$ and classify them. I'm fairly sure that due to the exponential term overpowering the factor $z$, there will be an essential singularity ...
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### Find $\int_{|z|=R} \frac{1}{(z-b)(z-a)^m} dz$

I have to find $\int_{|z|=R} \frac{1}{(z-b)(z-a)^m} dz$ for $|a| <R < |b|$ I would use Cauchy formula but first what can I do with $\frac{1}{(z-b)(z-a)^m}$? I dont remember it.
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### If |f| is constant, f is constant.

I am confused as to how they got from the two equations being equal to 0 to the derivative being 0. I could be really tired right now but this isn't really making sense to me. I was thinking of doing ...
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### Using Complex Analysis to Compute $\int_0 ^\infty \frac{dx}{x^{1/2}(x^2+1)}$

I am aware that there is a theorem which states that for $0<a<2$ we have $$\int_0^\infty\frac{x^{a-1}}{x^2+1}dx=\frac{\pi \cos\big(\frac{a\pi }{2}\big)}{\sin (a\pi) }$$ but I prefer to evaluate ...
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### Switching between Cartesian coordinate and polar coordinates

Under what assumption, every non-zero complex number represented in Cartesian coordinate system admits unique polar representation and vice versa ?
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### What are 'Regular Products'?

When looking at the functional equation for the Riemann zeta function, I came across the statement: For $s$ an even positive integer, the product $\sin{(\frac{\pi s}{2})}\Gamma({1-s})$ is regular. ...
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### When the argument of complex numbers is a well defined real valued function?

I know that the argument $\arg:\Bbb C\setminus\{0\}\to\Bbb R$ is multivalued function and also that if we consider $\arg:\Bbb C\setminus\{0\}\to{\Bbb R}/{2\pi \Bbb Z}$, then it is a well defined ...
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### Determine all points where $f(x+iy) = 2xy + i(x+\frac 2 3 y^3)$ is differentiable in $\mathbb C$.

Consider the function $f : \mathbb C \rightarrow \mathbb C$ given by $f(x+iy) = 2xy + i(x+\frac 2 3 y^3)$. I want to determine all points at which $f$ is differentiable as a complex function. To do ...
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### How to prove $\oint_\Gamma \nabla\theta\cdot\vec{dr}=\pm2\pi$ around a phase singularity/over a cut

How would you prove that $$\oint_\Gamma \nabla\theta\cdot\vec{dr}=\pm2\pi$$ We know that $\theta\in(-\pi,\pi)$, suppose that $\theta$ is continuous in the region bounded by and along $\Gamma$ apart ...
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### Divisor of the meromorphic differential $\omega=\frac{dx}{y^3}$ on C: $\xi_1^4+\xi_2^4=\xi_0^4$

Consider Fermat's curve of degree 4 defined by C : $\xi_1^4+\xi_2^4=\xi_0^4$ in projective coordinates $(\xi_0 :\xi_1 :\xi_2)$ or, equivalently, by the affine equation $x^4 + y^4 = 1$ in the affine ...
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### Show that the function $g(z) = f(e ^z )$ is not a polynomial.

Let $f :\mathbb C \rightarrow \mathbb C$ be an entire function. Show that the function $g(z) = f(e ^z )$ is not a polynomial. What is the technique to show a function is a polynomial?
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### How do I use residue theorem to evaluate this improper integral to get a good looking solution?

The problem is $\int_{0}^{\infty} \frac{\sqrt{x}}{x^2+2x+5}dx$ I replace x with z, and did some algebra, but the solution was rather nasty. it contains exponential and arctan such and such. However, ...
Now I thought it wouldn't be too much of an issue, but it is becoming hell to find the zeroes of: $$z^4 + 10z^2 +1$$ Now reason I need them is for the poles of a function I am working on. So with ...