Problem of complex equations and Cauchy-Riemann

Question

I have the following problem:

$$u (x, y) = \sin x \sinh y.$$

I need to check that the function is harmonic in the whole plane. But I do not know where to start, I would appreciate your help. Thanks for your attention.

Answer 1

Hint: The function is infinitely often continuously differentiable (has to be at least twice). Then check if the Laplace equation $\Delta u =0$ is fulfilled.

Answer 2

Note that $\cos(z)$ is analytic and $\cos(x+iy) = \cos(x) \cosh(y) - i \sin(x) \sinh(y)$. As $-$ the imaginary part of an analytic function, $\sin(x) \sinh(y)$ is harmonic.

Answer 3

Let $f(z)=i\cos(z)$. So, if $x,y\in\mathbb R$,\begin{align}f(x+yi)&=i\cos(x+yi)\\&=i\bigl(\cos(x)\cos(yi)-\sin(x)\sin(yi)\bigr)\\&=\sin(x)\sinh(y)+\cos(x)\cosh(y)i.\end{align}So, your function is the real part of an analytic function. Therefore, it is harmonic.