Question

a) Find the real part u(x,y) and imaginary part v(x,y) of f(2)= (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

2 ser: - Given function fra) (1+2i) z + (1-1) + 3 z=xting or fez) = (1+21) (x+y)+(-1) (x+y)+3 = (1+2i) (x yt + i 2xy) ti-) (x+y)+3 (x²y) ti izrey + 4. 2 (2013) - 4 xy +ix - y - xriy + 3 x-y + 3) +i (20-2y + 2xy y ) fer) = 4 (24) tiv(ay) the real part of f12) upring) = x2_y? - of xy - ren ny +3 and imaginary part v (kin) = 2*²-2y2 + 2xy tay Let Then by Now Now Un = 2x - 4y - 1 x = 4x + 2y + 1 by -zy - Anal - 4y + 2x -1 vy Now we Lave to verify the canchy- Riemann equations (CR equations) and y = -√ Here wy Use = 2n-ay - - 24 - 92-1 = -Vx and My Hence u and satis files the C-R egnations

Also u, v for all vel vy all Un, Cly, prints are continnen Herce given function is analytic function c) Now Unse = 2 aand Uyy = -2 Now we have the Laplace's equation Unntlyy Now Unn + Uyy @ 2-2 ie u (ny) satisfies the Laplace equation Horce Usa,u) is harmonic function. (Aus)