Timeline for Improper integral $\int^\infty_0 e^{-a x^2} \cosh (b\sqrt{1+x^2})$
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 21, 2014 at 19:05 | comment | added | Bazin | @user113103 The indicatrix function of $\mathbb R_+$ is the standard Heaviside function, which is 1 on the positive half-line and 0 on the negative half-line. $J_k(a)$ appears essentially as the $k$-th derivative of a product: just apply Leibniz formula. | |
| May 21, 2014 at 16:14 | comment | added | user113103 | As @TomDickens suggested in the comment below, Mathematica could evaluate $J_k(a)$. Mathematica gives $J_k(a)= \frac{1}{2} e^{-a} \left(a^{-k-\frac{1}{2}} \Gamma \left(k+\frac{1}{2}\right) \, _1F_1\left(-k;\frac{1}{2}-k;a\right)+\frac{\sqrt{\pi } \Gamma \left(-k-\frac{1}{2}\right) \, _1F_1\left(\frac{1}{2};k+\frac{3}{2};a\right)}{\Gamma (-k)}\right)$ where $_1F_1$ refers to confluent hypergeometric function of the first kind. | |
| May 21, 2014 at 16:05 | comment | added | user113103 | Thank you for the answer. Actually, it might be that your answer is explicit enough for my purpose, which is to do inverse laplace transformation of I(a,b). I don't follow the derivation completely, however. What is $1_{R_+}(x)$? Also, how $J_k(a)$ becomes a sum for $l$ is not clear to me. Could you explain a little bit more on this? | |
| May 21, 2014 at 15:53 | vote | accept | user113103 | ||
| May 20, 2014 at 20:40 | history | answered | Bazin | CC BY-SA 3.0 |