The principal argument of $\frac{1}{2+3\mathrm{i}}$ is _____.

The exponential form of $\mathrm{z}=\sqrt{3}-\mathrm{i}$ is _____.

One of the value of ${\left(-1+\sqrt{3} \mathrm{i}\right)}^{\frac{1}{3}}$ is _____.

The principal value of $\sin \left(\log {\mathrm{i}}^{\mathrm{i}}\right)$ is _____.

The value of $\underset{\mathrm{z}\to 0}{\lim }\frac{3{\mathrm{x}}^2+4{\mathrm{y}}^2}{{\left(\mathrm{x}+\mathrm{y}\right)}^2}$

The function $\mathrm{f}\left(\mathrm{z}\right)={\left(\mathrm{x}-\mathrm{y}\right)}^2+2\mathrm{i}\left(\mathrm{x}+\mathrm{y}\right)$ is analytic if $\mathrm{y}=$ _____ .

If $\mathrm{u}=2\mathrm{x}+{\mathrm{kx}}^3+3{\mathrm{xy}}^2$ is harmonic function, then $\mathrm{k}=$ _____.

One of the value of ${(1+\mathrm{i})}^{100}$ is _____ .

Let $\mathrm{z}=\mathrm{x}+\mathrm{iy} ; \mathrm{x}, \mathrm{y}\in \mathrm{R}$ and $\left|z-1\right| < \left|z+1\right|$ then _____ .

Let $\mathrm{z}=\mathrm{x}+\mathrm{iy} ; \mathrm{x}, \mathrm{y}\in \mathrm{R}$ . If $\left|\mathrm{i}+\mathrm{z}\right|=\left|\mathrm{i}-\mathrm{z}\right|$ , then _____ .