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 $t=0$.

  • 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 $\endgroup$ – AloneAndConfused Mar 10 '17 at 12:33

(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.

  • $\begingroup$ :Please mark some references for the expansion! The answer is very helpful! $\endgroup$ – Manu Mar 10 '17 at 16:21
  • $\begingroup$ It is implicitly contained e.g. in Abramovitz-Stegun. Or explicitly here. $\endgroup$ – Fabian Mar 10 '17 at 17:10

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