Plotting odd complex functions w/o computer help

Question

I am evaluating a complex integral that utilises the Cauchy Integral Formula and its properties.

In the book I'm reading, they give examples of evaluating integrals using CIT by graphing them, which really does help to see if points of a function is analytic in a certain domain.

For example, in evaluating this integral - how would I be able to plot the function shown in this integral, where C is a circle $|z|=1$ traversed once counter clockwise. $$\int_{C}\frac{z+i}{z^3+2z^2}$$

Thanks.

Answer 1

The idea is that you only need to know where the function is singular

In your case it is at $z= 0$ and $z = -2$, but the second point is outside the region of integration, so we do not care about it

$$ \oint_C \frac{z + i}{z^2(z + 2)} {\rm d}z= \oint_C \frac{f(z)}{z^2}{\rm d}z $$

where the function

$$ f(z) = \frac{z + i}{z + 2} $$

is well behaved in the region $|z| \leq 1$. The problem is at $z = 0$, where you have second order singularity, so you can use Cauchy's integral form

$$ \left.\frac{{\rm d}^n f}{{\rm d}z^n}\right|_{z = 0} = \frac{n!}{2\pi i}\oint_C \frac{f(z)}{z^{n + 1}}{\rm d}z $$

with $n = 1$ which yields

$$ \oint_C\frac{z + i}{z^2(z + 2)} = 2 \pi i \frac{{\rm d}}{{\rm d}z}\left(\frac{z + i}{z + 2}\right)_{z = 0} = 2\pi i \left(\frac{1}{2} - \frac{i}{4} \right) $$

Answer 2

Sketch: Note that the pole at $z=0$ has order $2$. Use the slightly generalized Cauchy formula for derivatives, $$f^{(n)}(a)=\frac{n!}{2\pi i}\oint \frac{f(z)}{(z-a)^{n+1}} \mathrm{d}z,$$ Write the original integrand as $g(z)$ and identify $f(z)$ from $g(z)=f(z)/z^2$ in your case for $n=1$ and $a=0$. Then compute the first derivative of $f$ at $a=0$ to compute the original integral (remember to multiply this by $2\pi i$). This yields $\pi (\frac12 +i)$.

Be sure to check all hypotheses of the Cauchy integral formula though to justify this rigorously, though (that $f$ is holomorphic inside the curve is required, not $g$, etc). Comment if you want some steps expanded, but preferably with where you got stuck also, or if you spot any typos, etc. It has been some time since I've gotten my hands dirty with contour integrals with only Cauchy's integral formula, however, the example on wikipedia is rather illustrative, though the poles there are only of order 1, unlike here.