Question

Harmonic conjugate of u(x, y)- e' cosx is

Answer 1

du/dx =e^x cos(y), d2u/dx2=e^x cos(y)

du/dy=-e^x sin(y), d2u/dy2= -e^x cos(y)

d2udx2+d2udy2=e^x cos(y)+ (-e^x cos(y))

d2udx2+d2udy2=0. .. U is the harmonic function

Ux=Vy

.:Vy=e^x cos(y).

f(z)=u+iv=e^x cos(y)+i(e^x cos(y))=e^z+c

f(z)=e(x+iy)=e^x*eiy=e^x(

cosy+icos(y))=e^xcosy+i

e^xcos(y).