New posts in Complex Analysis

calculate: $\int_{-\infty}^{\infty}\frac{\cos\frac{\pi}{2}x}{1-x^{2}}dx$ using complex analysis ; detect my mistake

Show that there does not exist a holomorphic function $h(z)$ such that $\exp(h(z)) = z$ on the punctured plane without using complex integration?

Show $1+\frac{8q}{1-q}+\frac{16q^2}{1+q^2}+\frac{24q^3}{1-q^3}+\dots=1+\frac{8q}{(1-q)^2}+\frac{8q^2}{(1+q^2)^2}+\frac{8q^3}{(1-q^3)^2}+\dots$.

Prove $ \int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$.

Prove $\lim_{z \to 0} \frac{z}{\overline{z}}$ doesn't exist using $\varepsilon-\delta$.

Prove: $\int_0^{\infty} \frac{\ln{(1+x)}\arctan{(\sqrt{x})}}{4+x^2} \, \mathrm{d}x = \frac{\pi}{2} \arctan{\left(\frac{1}{2}\right)} \ln{5}$

Convergence/Divergence of Complex Series $\sum\limits_{n=1}^{\infty} \frac{n(2+i)^n}{2^n}$

When does equality hold in $\Bigr\lvert\sum_{k=1}^n a_kb_k\Bigr\rvert^2 \le \left(\sum_{k=1}^n |a_k|^2\right)\left(\sum_{k=1}^n |b_k|^2\right)$?

Are all complex functions onto?

Proving $\frac1{2\pi} \int_0^{2\pi} \frac{R^2-r^2}{R^2-2Rr\cos\theta+r^2} d\theta =1$ by integrating $\frac{R+z}{z(R-z)}$ without residue theorem.