Question

Show that the real and imaginary parts of the function w=log z satisfies the Cauchy-Riemann equations when z is non zero.

Answer 1

choose the principle branch ... it's not difficult to show that the following function satisfies CR equations
\[ \log (x+iy) = \log (x^2 +y^2) + i \arctan(y/x) = u(x,y) + i v(x,y)\]

Answer 2

Is this the solution?Please show me the working.

Answer 3

Sorry, there is sqrt(x^2+y^2)
\[\log (x+iy) = \log (\sqrt{x^2 +y^2}) + i \arctan(y/x) = u(x,y) + i v(x,y) \]
You have to show that the following conditions hold
\[ \frac{\partial }{\partial x } \log (\sqrt{x^2 + y^2}) = \frac{\partial }{\partial y }\arctan(y.x) \\
\frac{\partial }{\partial y } \log (\sqrt{x^2 + y^2}) = -\frac{\partial }{\partial x }\arctan(y.x) \]
http://www.wolframalpha.com/input/?i=D%5BLog%5BSqrt%5Bx%5E2%2By%5E2%5D%5D%2C%7Bx%2C+1%7D%5D+%3D%3D+D%5BArcTan%5By%2Fx%5D%2C%7By%2C+1%7D%5D
http://www.wolframalpha.com/input/?i=D%5BLog%5BSqrt%5Bx%5E2%2By%5E2%5D%5D%2C%7By%2C+1%7D%5D+%3D%3D+-D%5BArcTan%5By%2Fx%5D%2C%7Bx%2C+1%7D%5D