# 复数函数代写|COMPLEX FUNCTIONS MATH243 University of Liverpool Assignment

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## Instructions:

Harmonic functions are a class of functions that arise naturally in many areas of mathematics and physics, including the study of partial differential equations, potential theory, and complex analysis.

In essence, a function is said to be harmonic if it satisfies Laplace’s equation, which is a partial differential equation that arises frequently in the study of physical systems. Laplace’s equation has many interesting properties, and solutions to this equation often exhibit beautiful symmetries and patterns.

Harmonic functions have many practical applications as well. For example, they are used in electrical engineering to model the flow of electric current through conductive materials, and they are also important in fluid dynamics, where they are used to study the behavior of fluids in motion.

Overall, the theory of harmonic functions is a rich and fascinating subject, with many connections to other areas of mathematics and physics. It is definitely worth exploring further if you are interested in these topics.

Find all solutions $z$ to equation $z^3=-8 i$.

Evaluate the integral
$$\int_{|z-1|=\frac{1}{2}} \frac{d z}{(1-z)^3} .$$

\begin{aligned} \int_{|z-1|=\frac{1}{2}} \frac{d z}{(1-z)^3} & =\int_0^{2 \pi} \frac{i e^i t / 2}{\left(-e^{i t} / 2\right)^3} d t \ & =\frac{-i}{2} \cdot 8 \int_0^{2 \pi} e^{-i \cdot 2 t} d t \ & =-\left.4 i \frac{e^{-2 i t}}{-2 i}\right|_0 ^{2 \pi} \ & =0 .\end{aligned}

Evaluate the integral
$$\int_\gamma \frac{e^z+z}{z-2} d z$$
in the two cases: 1) $\gamma={z:|z|=1}$;
2) $\gamma={z:|z|=3}$.

1) $\int_{|z|=1} \frac{e^2+z}{z-2} d z=0$, since $2 \notin{z:|z|<1}$.
2) $\int_{|z|=3} \frac{e^z+z}{z-2} d z=2 \pi i\left(e^2+2\right)$, since $n(\gamma, 2)=1$.