# 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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### Radial limits of composition of functions

Is it true that if $f\in H(U)$ is a holomorphic function whose nontangential limits exist a.e and $g\in H^\infty(U)$ is a nonconstant holomorphic function whose range is in $U$ and whose radial limits ...
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### Evaluate $\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$ using residue calculus

I'm asked to evaluate $$\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$$ $\mathbb{R}\ni a>0$, using residue calculus (where $\sqrt{\cdot}$ is the PV $\sqrt{}$). My approach is as follows: ...
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### Is the union of two projective curves in the projective plane a projective curve?

As the title suggests, is the union of two projective curves in the projective plane a projective curve? Any help would be appreciated, thanks.
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### Analytic variety is a countable union of complex manifolds

In an article on real analytic manifolds I came across the following remark: Let $W$ be a purely $k$-dimensional analytic subvariety of a domain in $\mathbb{C}^n$ and let $S$ be its singular locus. ...
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### A periodic entire function which must have a fixed point

I would like to check my work on the following problem: Suppose $f(z)$ is a non-constant periodic entire function satisfying $f(z+1)=f(z)$. Show that $f(z)$ has a fixed point. So my attempt is: ...
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### Show that the integral of Riemann function is analytic

I'm trying to resolve this problem. Let $\Omega$ be an open set no empty of $\mathbb C$, $[a,b]$ a compact interval of $\mathbb{R}$, further $f,\ g\colon[a,b] \to \mathbb C$ two integrable Riemann ...
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### Integrating $\int_{-\infty}^\infty \frac{1}{1 + x^4}dx$ with the residue theorem

Calculate integral $$\int\limits_{-\infty}^{\infty}\frac{1}{x^4+1} dx$$ with residue theorem. Can I evaluate $\frac 12\int_C \dfrac{1}{z^4+1} dz$ where $C$ is simple closed contour of the upper ...
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### Interpretation of the Argument Principle

Recall that the argument principle states that given a meromorphic function $f$ and a compact region $K \subseteq \mathbb{C}$ whose boundary determines a simple contour and on which $f$ has no ...
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### How do zeros on the complex plane affect the real number line?

Let's say there is a real-valued "signal" that you can only measure at discrete points $f(x)$. You have a theory that this signal is the result of an analytic function $f(z)$ on the complex plane but, ...
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### $\int_{-\infty}^{+\infty}\left|f(g(t))-f(h(g(t)))\right| \,dt=0$ How to find $f$?

Let $h,g$ be given entire functions. Consider $$\int_{-\infty}^{+\infty}|f(g(t))-f(h(g(t)))|\, dt=0,$$ where $|\cdot|$ means modulus. How do I find non-polynomial analytic solutions for $f\,$? ...
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### Is there a closed form expression for this integral?

I've been trying to find a closed form expression/series expansion for the following integral without success: $$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)dz$$...
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### Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$\theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) ,$$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
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### What is THE domain of analyticity of a holomorphic function?

I am self-studying complex analysis, and I am a little bit confused on notations. Suppose that $f:U \to \mathbb C$ is a holomorphic function defined on an open subset of $\mathbb C^n$. I understand ...