I need to evaluate the definite integral as a function of $t$, say find
$$\phi(t) = \int _0^{\infty} \exp\left(-\sqrt{x^2+t^2}\right)dx$$
and check continuity and differentiablity at $t=0$.
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2$\begingroup$ Since $e^{-(x^{2}+t^{2})}=e^{-x^{2}}\cdot e^{-t^{2}}$, you can factor out the $t$-dependence and calculate the integral $\endgroup$– AloneAndConfusedMar 10, 2017 at 12:33
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1 Answer
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(note that continuity and differentiability at $t=0$ can be discussed without evaluating the integral)
Assume $t>0$, after the substitution $x= t \sinh \xi$, we obtain the result $$\phi(t) = t\int_0^\infty\!d\xi\,e^{-t \cosh \xi} \cosh\xi. $$
This is a well-known representation of the modified Bessel function of the second kind. We obtain $$\phi(t) =t K_1(t) .$$
You can obtain the answers to your questions from the expansion (valid for small $t$) $$ K_1(t) = \frac1t + \frac14 t \left( \log\left(\frac{t^2}4\right) +2 \gamma -1\right) + O(t^3)$$ with $\gamma$ the Euler–Mascheroni constant.
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$\begingroup$ :Please mark some references for the expansion! The answer is very helpful! $\endgroup$– manuMar 10, 2017 at 16:21
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