Question

Determine whether the function u = e x cos y is harmonic or not.

Answer 1

∂u/∂x =e^x cos(y) , ∂2u/∂x2=e^x cos(y)

∂u/∂y= -e^x sin(y) , ∂2u/∂y2= -e^x cos(y)

∂2u∂x2+∂2u∂y2=e^x cos(y)+ (-e^x cos(y))

∂2u∂x2+∂2u∂y2=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).