is positive on $(0, \infty)$ with the finite limit $1$ as $\lambda \searrow 0$, and extends to an analytic function on ${\mathbb C}\setminus (-\infty, 0]$ if we let $\sqrt{\lambda}$ denote the principal branch of the square root. It also maps ${\mathbb C}_+$ to ${\mathbb C}_+$. Indeed, a straightforward calculation shows that $$\begin{aligned}
\operatorname{Im}\left(\frac{z}{\tanh z}\right)&=\frac{2}{|\exp(z)-\exp(-z)|^2} (\operatorname{Im}z \sinh(2\operatorname{Re}z)-\operatorname{Re}z \sin(2\operatorname{Im}z))\\
&> 
\frac{4}{|\exp(z)-\exp(-z)|^2} (\operatorname{Im}z \operatorname{Re}z-\operatorname{Re}z \operatorname{Im}z)=0\end{aligned}$$ when $\operatorname{Re}z, \operatorname{Im}z>0$, from which it follows that $
\operatorname{Im}(\sqrt{\lambda}/\tanh \sqrt{\lambda}))> 0$ when $\operatorname{Im}\lambda >0$. This implies that $\lambda \mapsto \tanh \sqrt{\lambda}/\sqrt{\lambda}$ satisfies the conditions of Theorem \[thm:Stieltjes extensions\]. Hence, $(g(\lambda))^{2\alpha}$ is a Stieltjes function by Lemma \[lemma:composition\].

Throughout the rest of Section \[sec:K\] we let $\alpha=1/2$. The following is our main result concerning the kernel $K$, and will be used repeatedly in the later sections.

\[prop:Whitham formula\] The Whitham kernel can be expressed in the form . The Borel measure $\sigma$ in the same formula satisfies $\int_{(0,\infty)}\frac{{{\mathrm d}}\sigma(t)}{t}<\infty$, and is absolutely continuous with density $$\frac{1}{\pi} \sum_{n=1}^\infty  \sqrt{\frac{|\tan \sqrt{t}|}{\sqrt{t}}}\chi_{((\frac{(2n-1)\pi}{2})^2, (n\pi)^2)}(t).$$ Thus $$\label{eq:explicit expression for Whitham kernel}
K(x)=\frac1{\pi} \sum_{n=1}^\infty\int_{\frac{(2n-1)\pi}{2}}^{n\pi} \exp(-s |x|)\sqrt{\frac{|\tan s|}{s}}\, {{\mathrm d}}s,$$ and $K$ is completely monotone on $(0,\infty)$. In particular, it is positive, strictly decreasing and strictly convex for $x>0$.

For $g$ as in we have 