One of the value of ${\left(\cos \frac{7\mathrm{\pi }}{8}+ i \sin \frac{7\mathrm{\pi }}{8}\right)}^{\frac{16}{7}}$ is equals to _____ .

Which of the following is/are true for analytic function $\mathrm{f}\left(\mathrm{z}\right)=\mathrm{u}\left(\mathrm{x}, \mathrm{y}\right)+ \mathrm{i} \mathrm{v}\left(\mathrm{x}, \mathrm{y}\right)?$

Which of the following statement is/are true?

I. There is no exist a function $\mathrm{u}\left(\mathrm{x}, \mathrm{y}\right)={\mathrm{x}}^2-{\mathrm{y}}^2$ such that $\mathrm{f}\left(\mathrm{z}\right)=\mathrm{u}\left(\mathrm{x}, \mathrm{y}\right)+ \mathrm{i} \mathrm{v}\left(\mathrm{x}, \mathrm{y}\right)$ is analytic.

II. There is no exist a function $\mathrm{u}\left(\mathrm{x}, \mathrm{y}\right)={\mathrm{x}}^2+{\mathrm{y}}^2$ such that $\mathrm{f}\left(\mathrm{z}\right)=\mathrm{u}\left(\mathrm{x}, \mathrm{y}\right)+ \mathrm{i} \mathrm{v}\left(\mathrm{x}, \mathrm{y}\right)$ is analytic.

The value of $\sum _{n=0}^{\infty } (\cos n\mathrm{\pi } + \mathrm{i} \sin \mathrm{n\pi }) =$ ____ .

Which of the following are fixed points of $w=\frac{2z+6}{z+7}?$

At which points $\mathrm{w}=\mathrm{z}+\frac{1}{\mathrm{z}}$ is not confoemal?

Let $\mathrm{w}=\frac{\mathrm{pz}+\mathrm{q}}{\mathrm{az}+\mathrm{b}}$ , then w is mobius transformation if _____ .

The transformation $\mathrm{w}=\frac{1}{\mathrm{z}}$ maps $\mid z\mid <1$ on to ______ .

The image of $0<y<2$ under $\mathrm{w}=\mathrm{i} \mathrm{z}+1$ , where $\mathrm{z}=\mathrm{x}+\mathrm{iy}, \mathrm{w}=\mathrm{u}+\mathrm{iv} \& \mathrm{i}=\sqrt{-1}$ , is _____ .

The function $\mathrm{w}={\mathrm{z}}^2-\mathrm{z}$ is conformal everywhere except at $z=\_\_\_\_\_ .$