List of top Mathematics Questions on Complex Functions

For a given \[ u(x,y)=x^3-3xy^2+3x^2-3y^2+1, \] the analytic function \[ f(z)=u+iv \] is

For a given \[ u(x,y)=x^3-3xy^2+3x^2-3y^2+1, \] the analytic function \[ f(z)=u+iv \] is

The Laurent's series for the function \( f(z) = 1 + \frac{3{z+2} - \frac{8}{z+3} \) in the region \( |z| < 2 \) is}

Let $r = \text{Min}\{\alpha, \beta, \gamma\}$, $R = \text{Max}\{\alpha, \beta, \gamma\}$, $f(z) = \frac{z}{(z-\alpha)(z-\beta)(z-\gamma)}$. $I_1 = \oint_{C_1} f(z)dz$ and $I_2 = \oint_{C_2} f(z)dz$, where $C_1 : |z| < r$ and $C_2 : |z| = R+1$, then $I_1 + I_2 = $}

Given \[ P(A)=\frac{7}{20}, \quad P(B)=\frac{1}{2}, \quad P(C)=\frac{9}{20}, \] \[ P(A\cap B\cap C)=\frac{1}{20}, \quad P(A\cap B)=\frac{3}{20}, \] \[ P(A\cap C)=\frac{1}{8}, \quad P(B\cap C)=\frac{1}{5}, \] then \(P(B|\overline{A})\) is

Evaluate \[ \oint_C \frac{z+2}{z^2+2z+2}\,dz, \] where \(C\) is the circle \[ |z-\alpha|=1, \qquad \alpha=-1+\frac{3}{2}i. \]