Hi there,

With z = u(x,y) + iv(x,y) I have to show that exp|z| is not analytic. so I have

exp|z| = exp{(u^2 + v^2)^1/2}

as |z| is sqrt(u^2 + v^2).

I know that showing the Cauchy Riemann functions aren't satisfied will prove it's not analytic, but I'm not sure how to introduce this.

If \(\displaystyle z = u(x, y) + i\,v(x, y)\)

then \(\displaystyle |z| = \sqrt{[u(x, y)]^2 + [v(x, y)]^2}\).

This is a REAL function, not complex.

Therefore \(\displaystyle e^{|z|} = e^{\sqrt{[u(x, y)]^2 + [v(x, y)]^2}}\) is also real.

So we can write it as

\(\displaystyle Z = e^{\sqrt{[u(x, y)]^2 + [v(x, y)]^2}} + 0i = U + i\,V\).

You should be able to see that all the partial derivatives of \(\displaystyle V\) will be \(\displaystyle 0\), while the partial derivatives of \(\displaystyle U\) won't always be.

Therefore the Cauchy-Riemann equations will not be satisfied, and the function is not analytic.