I am attempting to show that $\sinh(\mathrm{e}^z)$, where $z$ is a complex number, is entire. The instructions of the problem tell me to write the real component of this function as a function of $x$ and $y$, which I used algebra to do; this function is $u(x, y)=\cos(\mathrm{e}^x \sin(y))\cosh(\mathrm{e}^x \cos(y))$. The instructions then say to "state why this function must be harmonic everywhere", i.e., that the sum of the second derivatives $u_{xx}, u_{yy}$ is zero.
This is where I'm stuck: Even the first derivative looks like a nightmare to compute, and the language of the problem's instructions seems to suggest that I shouldn't need to, that I should simply be able to glance at the function and know that it's harmonic, and state why in a simple sentence. Why should this function be clearly harmonic?
Best Answer
You should have a theorem in your text to the effect of "the composition of two entire functions is entire" and "the sum of two entire functions is entire".
Note that if $\sinh(z)$ is entire, then $\sinh(e^z)$ must be entire since this would be the composition of entire functions.
Note that $$ \sinh(z) = \frac 12 \left(e^z - e^{-z}\right) $$ Is the sum of entire functions.