Question

Problem 2. (15 points) a) Find the real part u(x,y) and imaginary part v(x,y) of f(z) = (1+2i)z+ (i – 1)2 +3 b) Verify if the above function is analytic c) Using Laplace's equation verify if the real part u(x,y) is harmonic.

Answer 1

Problem 2.

Solution:

$\small a)$

f (2) = (1+21):2+ (i-1): +3

::f(-) = (1 +21) (x + y)+ (i-1) (x + iy) + 3

:. f () = (x2 - y2 - 4ry) +i (2x2 – 2y2 + 2xy) + (-1- y) +i (2-y) + 3

$\small \therefore f\left ( z \right )=\left ( x^{2}-y^{2}-4xy-x-y+3 \right )+i\left ( 2x^{2}-2y^{2}+2xy +x-y\right )$

$\small \therefore u\left ( x,y \right )+iv\left ( x,y \right )=\left ( x^{2}-y^{2}-4xy-x-y+3 \right )+i\left ( 2x^{2}-2y^{2}+2xy +x-y\right )$

$\small \therefore u\left ( x,y \right )=x^{2}-y^{2}-4xy-x-y+3,\: \: \: \: \: \: \: \: \: \: \: \: v\left ( x,y \right )= 2x^{2}-2y^{2}+2xy +x-y$

Thus , real part $\small u\left ( x,y \right )=x^{2}-y^{2}-4xy-x-y+3$ and the imaginary part

$\small v\left ( x,y \right )= 2x^{2}-2y^{2}+2xy +x-y$

$\small b)$

$\small u\left ( x,y \right )=x^{2}-y^{2}-4xy-x-y+3,\: \: \: \: \: \: \: \: \: \: \: \: v\left ( x,y \right )= 2x^{2}-2y^{2}+2xy +x-y$

$\small \therefore u_{x}=2x-4y-1,\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: v_{x}=4x+2y+1$

$\small u_{y}=-2y-4x-1,\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: v_{y}=-4y+2x-1$

Since $\small u_{x}=v_{y}$ and  $\small u_{y}=-v_{x}$ i.e. the Cauchy-Riemann equation is satisfield,  $\small f\left ( z \right )$ is an analytic function.

$\small c)$

$\small u_{x}=2x-4y-1$

$\small u_{xx}=2$

$\small u_{y}=-2y-4x-1$

$\small u_{yy}=-2$

$\small \therefore u_{xx}+u_{yy}=2-2=0$

$\small \therefore u\left ( x,y \right )$ is harmonic.