Where's $\dfrac{\sinh(\sin z)}{z^2+9}$ analytic?

Question

The exercise pretty much asks to find the domain on which the function above, $$f(z)=\dfrac{\sinh(\sin z)}{z^2+9},$$ is analytic.

The numerator, $\sinh(\sin z)$, is an entire function so we shall only care about the denominator: we should impose $z^2+9\neq 0\iff z\neq\pm 3i$. Still, I'm pretty lost on how this relates to the domain of analycity since in the solutions they talk about singularities at $z=\pm i$ (most likely a typo?). Finally, does Cauchy-Riemann conditions intervene here? Would that provide some more information about the domain we want to find?

Answer 1

You're right. The only singularities are at $z = \pm 3i$, since a quotient of analytic functions is analytic unless the denominator vanishes.

The Cauchy-Riemann equations don't give any additional information here:

• Away from the singularities, you've already noticed that there's a more convenient way to see that the function is analytic; no need to go through the trouble of calculating the real and imaginary parts.
• At the singularities, the partial derivatives in the Cauchy-Riemann equations don't even exist, so the equations fail in a trivial way.

(The Cauchy-Riemann equations are only "interesting" when you have a function $\mathbb{R}^2 \to \mathbb{R}^2$ that's already real-differentiable at a point, and you're wondering if it's also complex-differentiable there. If the function is not even real-differentiable, there's nothing to check.)