# Partial derivative of the heat kernel

I happen to have the heat kernel on the two-dimensional hyperbolic space and I need to take partial derivatives in order to check that it satisfies the heat equation as expected. The problem is I can not apply the Leibniz formula because I get zero in the denominator. The function is $$P_2(x,t)=\frac{\sqrt{2}e^{-t/4}}{(4\pi t)^{3/2}}\int_x^\infty\frac{se^{-s^2/4t}ds}{\sqrt{\cosh(s)-\cosh(x)}}.$$ I would be very grateful if you could help me take the partial derivative $$\frac{\partial P_2}{\partial x}$$. I need it symbolically, not numerically, because I want to use it in the heat equation.

• How about substituting $s = r x$ and using the dominated convergence theorem? Nov 15, 2021 at 11:24

A partial integration can remove the singularity: $$P_2(x,t)=\frac{\sqrt{2}e^{-t/4}}{(4\pi t)^{3/2}}\int_x^\infty\frac{se^{-s^2/4t}ds}{\sqrt{\cosh s -\cosh x }}=$$ $$\qquad =\frac{\sqrt{2}e^{-t/4}}{(4\pi t)^{3/2}}\int_x^\infty\frac{2\sqrt{s-x}\,s e^{-s^2/4t}}{\sqrt{\cosh s -\cosh x}}\left(\frac{d}{ds}\sqrt{s-x}\right)\,ds$$ $$\qquad=-\frac{\sqrt{2}e^{-t/4}}{(4\pi t)^{3/2}}\int_x^\infty\sqrt{s-x}\left(\frac{d}{ds}\frac{2\sqrt{s-x}\,s e^{-s^2/4t}}{\sqrt{\cosh s -\cosh x}}\right)\,ds$$ $$\qquad=\frac{\sqrt{2}e^{-t/4}}{(4\pi t)^{3/2}}\int_x^\infty e^{-s^2/4 t}\frac{ \left(s^3-s^2 x-3 s t+2 t x\right) (\cosh s-\cosh x)+s t (s-x) \sinh s}{t (\cosh s-\cosh x)^{3/2}}\,ds.$$ In the final expression the integrand vanishes$$^\ast$$ as $$(s-x)^{1/2}$$ when $$s\rightarrow x$$, so there are no contributions from the integration bounds when we differentiate the integral with respect to $$x$$.
$$^\ast$$ The numerator expands around $$s=x$$ as $$\left(s^3-s^2 x-3 s t+2 t x\right) (\cosh s-\cosh x)+s t (s-x) \sinh s$$ $$\qquad=-xt(s-x)\sinh s+xt(s-x)\sinh s+{\cal O}(s-x)^2={\cal O}(s-x)^2.$$ The denominator is of order $$(s-x)^{3/2}$$, so the ratio is of order $$(s-x)^{1/2}$$.
• Thank you for your answer. However I fail to see why the integrand vanishes as $(s-x)^{1/2}$. Could you please give me more details on that? Nov 15, 2021 at 17:13