# Some basics questions on complex integration

I recently started studying complex function theory and I'm solving some exercises to understand the various theorems:

The firt exercise is:

Let $$R := \{z = x + i y : |x| <1 \text{ and } 0 . Determine (without calculation) the value of the integral $$\int_{∂R} \frac{1}{(z − i)^3}dz$$.

I have following solutions:

We have $$\int_{∂R} \frac{1}{(z − i)^3}dz=-1/2(z-i)^{-2}|_{∂R}=0$$

How did he understand so easily that the integral returned zero? What theorem did he use?

The second exercise where I have some questions is:

Determine the value of the integral $$\int_{|z|=2}\frac{sin(z)}{z-\pi/4}dz$$

The solutions are:

According to Cauchy's integral theorem

$$\int_{|z|=2}\frac{sin(z)}{z-\pi/4}dz=2\pi i \dot \sin(\pi/4)=\pi i \sqrt{2}$$

I don't understand how we manage to solve this integral using cauchy. Could someone explain it to me, perhaps by writing some middle steps?

• 0) Please only 1 question/post 1) Which of the two $=$ don't you understand in $\int_{∂R} \frac{1}{(z − i)^3}dz=-1/2(z-i)^{-2}|_{∂R}=0$? 2) Do you know Cauchy's integral theorem? If so, what stops you precisely? Commented Dec 14, 2023 at 13:58
• $-1/2(z-i)^{-2}|_{∂R}=0$ @AnneBauval . I know the theorem but I still find it difficult to apply it Commented Dec 14, 2023 at 14:07
• Only a $\frac{1}{z-i}$ term would make the integral nonzero. There isn't one of those.
– J.G.
Commented Dec 14, 2023 at 14:16
• 1) What does $f(z)|_\Gamma$ mean to you, for a curve $\Gamma$ parametrized by some function $\gamma:[0,1]\to\Bbb C$? 2) What is your statement of that theorem and where precisely is your difficulty in applying it? (Please, answer to both questions by editing your post, rather than by a comment.) Commented Dec 14, 2023 at 15:40

For your first question, to solve the integral $$\int_{\partial R}\frac{1}{(z-i)^3}\mathrm{d}z$$, and note $$z=i$$ is inside domain $$R$$, one may use Goursat's formula: $$f^{(n)}(a)=\frac{n!}{2\pi i}\int_{C}\frac{f(z)}{(z-a)^{n+1}}\mathrm{d}z$$ where $$C$$ is enclosed curve and $$a$$ inside $$C$$. Then $$a=i$$ and $$f(z)=1$$, then it's easily to see the value of $$\int_{\partial R}\frac{1}{(z-i)^3}\mathrm{d}z$$ is $$\frac{2\pi i}{2!}f^{(2)}(i)$$, it is zero.
Similarly, for second question, the point $$z=\frac{\pi}{4}$$ is inside circle $$|z|=2$$, the Cauchy's integral formula gives the answer. $$\int_{|z|=2}\frac{\sin(z)}{z-\frac{\pi}{4}}\mathrm{d}z=2\pi i\cdot \sin(\frac{\pi}{4})$$