Is it possible to differentiate a modified bessel function of the third kind?

Question

Suppose that we have two variables, $\alpha\in \mathbb{R}^+$ and $\beta\in \mathbb{R}^+$. Then we have the following modified Bessel function of the third kind, $$\delta = K_{1} \left(\sqrt{\alpha\beta}\right). $$ I am interested to find both $\frac{\partial \delta}{\partial \alpha}$ and $\frac{\partial \delta}{\partial \beta}$. Is it possible to obtain such derivatives?

Answer 1

Bessel $K_n(x)$ is an analytic function of $x$ at all points of $\mathbb R^+$ for all $n$. So it is certainly differentiable, and you can find $\partial \delta/\partial \alpha$ via the chain rule: $$ \frac{\partial \delta}{\partial \alpha} = \frac{K_1'(\sqrt{\alpha\beta})}{2}\sqrt{\frac{\beta}{\alpha}} = -\frac{K_0(\sqrt{ab})+K_2(\sqrt{ab})}{4}\sqrt{\frac{\beta}{\alpha}} $$