Solved Problem on Cauchy's Integral Formula
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e) \( \displaystyle \oint_{{|z|=1}}\frac{z^{2}+z+\mathrm{i}}{(4z-\mathrm{i})^{3}}\;dz \)
Writing the integral as
\[
\begin{gather}
\oint_{|z|=1}\frac{z^{2}+z+\mathrm{i}}{\left[4\left(z-\dfrac{\mathrm{i}}{4}\right)\right]^{3}}\;dz\\[5pt]
\oint_{|z|=1}\frac{z^{2}+z+\mathrm{i}}{64\left(z-\dfrac{\mathrm{i}}{4}\right)^{3}}\;dz
\end{gather}
\]
The path of integration is given by the circle of radius 1, centered at the origin (0, 0), traversed
counterclockwise (Figure 1).
\[
\begin{gather}
\bbox[#99CCFF,10px]
{f^{(n)}(z_{0})=\frac{{n}!}{2\pi \mathrm{i}}\;\oint_{{C}}\frac{f(z)}{\left(z-z_{0}\right)^{n+1}}\;dz} \tag{I}
\end{gather}
\]
Identifying the terms of the integral
\[
\begin{gather}
\frac{{
\bbox[#FFCC66,2px]
{n}
}!}{2\pi \mathrm{i}}\;\oint_{{C}}\frac{
\bbox[#FFFF66,2px]
{f(z)}
}{\left(z-
\bbox[#FFD9CC,2px]
{z_{0}}
\right)^{
\bbox[#FFCC66,2px]
{n}
+1}}\;dz=\oint_{{|z|=1}}
\bbox[#FFFF66,2px]
{{\frac{z^{2}+z+\mathrm{i}}{64}}}
\frac{1}{\left(z-
\bbox[#FFD9CC,2px]
{{\dfrac{\mathrm{i}}{4}}}
\right)^{
\bbox[#FFCC66,2px]
{2}
+1}}\;dz
\end{gather}
\]
the point \( z-\frac{\mathrm{i}}{4}=0\Rightarrow z=\frac{\mathrm{i}}{4} \), lies inside the region determined by de closed contour C, will be used in the calculation of the integral we have \( f(z)=\frac{z^{2}+z+\mathrm{i}}{64} \), \( z_{0}=\frac{\mathrm{i}}{4} \), and n = 1, writing the expression (I)
\[
\begin{gather}
\oint_{{C}}\frac{f(z)}{\left(z-z_{0}\right)^{n+1}}\;dz=\frac{2\pi\mathrm{i}}{n!}\;f^{(n)}(z_{0})\\[5pt]
\oint_{{|z|=1}}\frac{z^{2}+z+\mathrm{i}}{(4z-\mathrm{i})^{3}}\;dz=\frac{2\pi\mathrm{i}}{2!}\;f^{(2)}\left(\frac{\mathrm{i}}{4}\right)
\end{gather}
\]
Calculation of the second derivative of
\( \displaystyle f(z)=\frac{z^{2}+z+\mathrm{i}}{64} \)
\[
\begin{gather}
\frac{df}{dz}=\frac{2z+1}{64}\\[5pt]
\frac{d^{2}f}{dz^{2}}=\frac{2}{64}=\frac{1}{32}
\end{gather}
\]
\[
\begin{gather}
\oint_{|z|=1}\frac{z^{2}+z+\mathrm{i}}{(4z-\mathrm{i})^{3}}\;dz=\frac{2\pi\mathrm{i}}{2\times 1}\;f^{(2)}\left(\frac{\mathrm{i}}{4}\right)\\[5pt]
\oint_{|z|=1}\frac{z^{2}+z+\mathrm{i}}{(4z-\mathrm{i})^{3}}\;dz=\pi\mathrm{i}\frac{1}{32}
\end{gather}
\]
\[
\begin{gather}
\bbox[#FFCCCC,10px]
{\oint_{{|z|=1}}\frac{z^{2}+z+\mathrm{i}}{(4z-\mathrm{i})^{3}}\;dz=\frac{\pi\mathrm{i}}{32}}
\end{gather}
\]
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .