Steps to find the conjugate harmonic function and the analytic function in Complex Variables

Steps to find the conjugate harmonic function and the analytic function in Complex Variables

Topic

Complex Variables – Analytic Function
Type 1 – Steps to find the derivative of an analytic function in Complex Variables
Type-2 – Steps for construction of analytic function given its real or imaginary part in complex variables
Type-3 – Steps to find the conjugate harmonic function and the analytic function in Complex Variables
Type 4 – Steps for Analytic Functions not Categorized in any types in Complex Variables

Type 1 Analytic function problems

1. Show that f(z)=sin z is analytic and hence find f’(z)
2. Show that f (z) =cos hz is analytic and hence find f’(z)
3. Show that f (z) =z^n where n is a positive integer is analytic and hence find its derivative
4. Show that w=z z(bar) is not analytic function in complex variables
5. Show that f (z) =e^x (cos y + i sin y) is analytic function in complex variable
6. Find the derivative of the analytic function z^z
7. Given that f(z)= (x^3 (1+i )-y^3 (1-i))/(x^2+y^2 ) if z≠0 and f(z) =0 if z=0, Show that f(z) satisfies Cauchy-Riemann equations at the origin
8. Show that f (z)=(r+k^2/r cosθ+i (r-k^2/r) sinθ, r≠0 is a regular function of z=r e^ (iθ) Also find f'(z)

Type 2 Analytic function problems

1. Construct the analytic function whose real part is u=log(sqrt((x^2+y^2))
2. Find the analytic function f (z) whose imaginary part is e^x (x siny + y cosy)
3. Find the analytic function f (z) given u=e^ (-x) [(x^2-y^2) cosy+2xy siny]
4. Determine the analytic function f (z) =u+iv given that the real part u=e^2x (x cos(2y-y sin2y))
5. Construct the analytic function whose real part is r^2 cos 2theta

Type 3 Finding the conjugate harmonic function and the analytic function

1. Show that u=x^3-3xy^2+3x^2-3y^2+1 is harmonic and find its harmonic conjugate. Also find the corresponding analytic function f (z)
2. Determine which of the following function is harmonic and find the conjugate harmonic function and express u+iv as an analytic function of z.(a) v=log sqrt(x+y) (b) v=cos x sinhy
3. Show that u=e^x (x cosy-y siny) is harmonic and find its harmonic conjugate. Also determine the corresponding analytic function
4. Given that u=x^2+4x-y^2+2y as the real part of an analytic function, find v and hence find f (z) in terms of z
5. Show that u=(r+1/r) cos(theta) is harmonic. Find its harmonic conjugate and also the corresponding analytic function

Type-4: Miscellaneous Problems

1. Find the analytic function f (z) =u+iv given u-v=e^x (cosy-siny)
2. Find the analytic function f (z) as a function of z given that the sum of its real and imaginary parts is x^3-y^3+3xy(x-y)
3. If f (z) =u+iv is analytic, find f (z) if u-v=(x-y) (x^2+4xy+y^2)
4. If f (z) =u(r, θ) +iv(r, θ) is analytic and given that u+v=1/r^2 (cos(2theta-sin2theta), r≠0, determine the analytic function f (z)
5. If u and v are harmonic functions show that (del u/del y-del v/del x) +i (del u/del y+ del v/del y) is analytic
6. If f(z)is analytic, show that [del^2/(delx^2 )+del^2/(dely^2 )] mod(f(z))^2=4mod(f'(z))^2
7. If f(z) is a regular function of z, show that {(del/delx) modf(z)}^2+ {(del/dely)modf(z)}^2= modf'(z)^2
8. If f (z) is analytic show that log mod(f (z)) is harmonic