Complex Variables and Partial Differential Equations (3130005)

Question-3b-OR

BE | Semester-3 Winter-2019 | 26-11-2019

Q3) (b)

Let,

f (z) = u + iv

, is an analytic function.

\Rightarrow Re (f (z)) = u \Rightarrow \Rightarrow {|Re (f (z))|}^{2} = u^{2}

∎ \frac{\partial}{\partial x} {|Re (f (z))|}^{2} = \frac{\partial}{\partial x} u^{2} = 2 u u_{x}

\Rightarrow \frac{\partial^{2}}{\partial x^{2}} {|Re (f (z))|}^{2} = \frac{\partial^{2}}{\partial x^{2}} u^{2} = 2 (u u_{xx} + u_{x} u_{x}) = 2 {uu}_{xx} + u_{x}^{2} - - - - - (1)

∎ \frac{\partial}{\partial y} {|Re (f (z))|}^{2} = \frac{\partial}{\partial y} u^{2} = 2 u u_{y}

\Rightarrow \frac{\partial^{2}}{\partial y^{2}} {|Re (f (z))|}^{2} = \frac{\partial^{2}}{\partial y^{2}} u^{2} = 2 (u u_{yy} + u_{y} u_{y}) = 2 {uu}_{yy} + u_{y}^{2} - - - - - (2)

Taking

[1] + [2]

, We have

(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) {|Re (f (z))|}^{2} = 2 {uu}_{xx} + u_{x}^{2} + 2 {uu}_{yy} + u_{y}^{2}

(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) {|Re (f (z))|}^{2} = 2 u (u_{xx} + u_{yy}) + 2 (u_{x}^{2} + u_{y}^{2})

(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) {|Re (f (z))|}^{2} = 2 {|f^{'} (z)|}^{2}

Because

(1) f (z) = u + iv

is an analytic function, then u is a harmonic function. So,

u_{xx} + u_{yy} = 0

(2) f (z) = u + iv \Rightarrow f^{'} (z) = u_{x} + {iv}_{x} = u_{x} - {iu}_{y}

by CR - equations.

(2) f (z) = u + iv \Rightarrow |f^{'} (z)| = \sqrt{(u_{x}^{2} + u_{y}^{2})}

(2) f (z) = u + iv \Rightarrow {|f^{'} (z)|}^{2} = (u_{x}^{2} + u_{y}^{2})