Finding an antiderivative of complex function $\sin(z^2)$

Question

I am trying to find the antiderivative of the complex function $\sin(z^2)$ as well as $\sin(z^2)/(z-i) $ and $\sin(z^2)/(z-i)^2$. The only hint don't think profoundly. Also I have to evaluate the integrals of the above three functions respectively on the unit circle the circle centered at $i$ of radius 2 and the circle centered at $i$ of radius 4.

Answer 1

The notion of anti-derivatives in Complex Analysis is tied to the presence of some path $\gamma$ of a path integral. That is, in Calculus of Real variables, it's ok to write $\int \sin(x^2)dx$ and think about anti-derivatives as an indefinite integral.

But in Complex, we don't (usually) think about $\int \sin(z^2)dz$ without considering some path or contour associated with the latter integral. See Antiderivatives on Wikipedia or Marsden and Hoffman, page 117.

Now, let $\gamma_1$ be the unit circle, $\gamma_2$ be the circle centered at $i$ of radius 2, and $\gamma_3$ be the circle centered at $i$ of radius 4. (Do we get to assume the orientation of each of these $\gamma$s is counter-clockwise?)

1. For $\displaystyle \int_{\gamma_1} \sin(z^2) dz$, what can you say about the $\sin(z^2)$ function inside $\gamma_1$? Are there any singularities inside $\gamma_1$?

If we find there are a finite number of singularities, then $\sin(z^2)$ may be meromorphic (in a simply connected domain $\mathbb{C}$) - in which case we may apply the Residue Theorem to evaluate the integral. Then

However, if there are no singularities, there may be another theorem you may apply which makes this integral significantly easier. If you don't know the name of the theorem, it's named after this guy.

2. For $\displaystyle \int_{\gamma_2} \dfrac{\sin(z^2)}{(z-i)}$, let $f(z) = \dfrac{\sin(z^2)}{(z-i)}$ and go through the same process (ask yourself the same questions). Are there any singularities inside $\gamma_2$? If so, how would you classify them? We need to classify the singularity in order to know which formula to use to calculate the singularity's residue. We'll answer this question partially to guide you to the correct answer.

Note at $z = i$ we have a simple pole. This is the only singularity inside $\gamma_2$. Since $f(z)$ is meromorphic in $\mathbb{C}$ (a simply connected domain) and $\gamma_2$ is a closed curve, we may apply the Residue Theorem to obtain

$\displaystyle \int_{\gamma_2} \dfrac{\sin(z^2)}{(z-i)} = 2\pi i \cdot$Res($f(z), i$)),

Where by classifying $z = i$ as a simple pole, we may find the "Residue at a simple pole" formula to evaluate the residue and complete the integral calculation. See your class notes or this link for residue formulas.

For $\displaystyle \int_{\gamma_3} \frac{\sin(z^2)}{(z-i)^2}$, you will have to repeat the process: analyze the types of singularities inside $\gamma_3$, see if you can apply the Residue Theorem (usually yes), apply the Residue Theorem, then calculate the residue based off the classification of the singularity/singularities.