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3
votes
1answer
199 views
Complex antiderivative
I am confused on a couple things:
1.) Why is it that an integral of a complex valued function of a complex variable exists if f(z(t)) is piecewise continuous (and/or piecewise continuous on ...
4
votes
2answers
151 views
Show the existence of a complex differentiable function defined outside $|z|=4$ with derivative $\frac{z}{(z-1)(z-2)(z-3)}$
My attempt
I wrote the given function as a sum of rational functions (via partial fraction decomposition), namely
$$
\frac{z}{(z-1)(z-2)(z-3)} = \frac{1/2}{z-1} + \frac{-2}{z-2} + \frac{3/2}{z-3}.
...
2
votes
2answers
162 views
Evaluate $\int_{C}\frac{e^{\alpha z}}{z}dz$ where $\alpha \in \mathbb R$ and C is the circle $\gamma(t)=e^{it}$…
Let $\alpha \in \mathbb R$ and C be the circle $\gamma(t)=e^\alpha t$, $-\pi\le t \le \pi$
Evaluate $$\int_{C}\frac{e^{\alpha z}}{z}dz.$$
Use the above, to show that $$\int_{0}^{\pi}e^{\alpha ...
