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Veva Garcia edited untitled.tex
about 8 years ago
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\begin{problem}[4.34]
Use the Cauchy Integral Formula(Theorem 4.30) to evaluate the integral in Exercise 4.33 when r=3.
\begin{proof}
Consider the integral
$\int_{c[0,3]} $$\int_{c[0,3]} \frac{1}{z^2-2z-8}dz=\int_{c[0,3]}
\frac{dz}{(z+2)(z-4)}$.\\ \frac{dz}{(z+2)(z-4)}$$.
Notice the function $f(z)=\frac{1}{z-4}$ is holomorphic in $\mathbb{C}$$\setminus\left\{4\right\}$ which contains $D\overline [-2,3]$. Thus, we can apply the Cauchy Integral Formula(Theorem 4.30):
\begin{align*}
\frac{1}{2\pi i}\int_{c[-2,3]}\frac{\frac{1}{z-4}}{z+2}dz\\