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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Subadditivity for Analytic Capacity Disjoint Compacts separated by a Line

The following problem is asked in Greene and Krantz, Problem 9, page 382: Suppose that $C_1$ and $C_2$ are disjoint compact sets in $\mathbb{C}$ that can be separated by a line $l$ with $C_1 \cap ...
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About the branch-cut in the complex logarithm

Say I have the function $log (f(z))$, then does the imaginary part of the value go down by $-i 2 \pi$ on crossing the branch-cut of the log function even if $f(z)$ is not crossing the branch-cut? ...
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second order linear ode in the complex domain

Consider $w''(z)+p(z)w'(z)+q(z)=0$ where $p(z), q(z)$ are analytic for $R\le|z|<\infty$ for some fixed $R$. Now I want to prove using analytic continuation of the solutions that the ode has one ...
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Formula for decomposing a form into $(p,q)$ forms

Let $L: \mathbb{C}^n \to \mathbb{C}$ be a real linear map. In other words, $L(a\vec{v}_1+b\vec{v_2}) = aL(\vec{v}_1)+bL(\vec{v}_2)$ for all $a,b \in \mathbb{R}$. Then $L$ decomposes uniquely into a ...
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Conditions required for $(z_{1}z_{2})^{\omega}=z_{1}^{\omega}z_{2}^{\omega}$, where $z_{1},z_{2},\omega\in\mathbb{C}$

I am having trouble finding the conditions on $z_{1}$ and $z_{2}$ in order for: $$(z_{1}z_{2})^{\omega}\equiv z_{1}^{\omega}z_{2}^{\omega}\qquad \forall\omega\in\mathbb{C}$$ My first step was to ...