Thread: Show that u is the real part of analytic function

Answer

u(x,y) is the real part of an analytic function if it has zero Laplacian. Note that

u_x = 2x
u_xx = 2

u_y = -2y - 1
u_yy = -2

Therefore, the Laplacian of u is

u_xx + u_yy = 2 - 2 = 0.

Is the answer correct.....

Originally Posted by mathsforumhelp

Answer

u(x,y) is the real part of an analytic function if it has zero Laplacian. Note that

u_x = 2x
u_xx = 2

u_y = -2y - 1
u_yy = -2

Therefore, the Laplacian of u is

u_xx + u_yy = 2 - 2 = 0.

Is the answer correct.....

Yes for both questions you need find the Laplacian and see if it is equal to zero. If it is the function is harmonic. Harmonic functions are the real and imagary parts complex functions.