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Results tagged with asymptotics
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user 101078
Asymptotic behavior of functions, asymptotic series and related topics
3
votes
0
answers
174
views
On analogues of Weber's formula
Let $J_0(x)$ be the $0$-th Bessel function of the first kind. Weber's formula states that
$$
\int_0^{+\infty}e^{-x}J_0(2\sqrt{\alpha x})J_0(2\sqrt{\beta x})dx=e^{-\alpha-\beta}I_0(2\sqrt{\alpha\beta}) …
- 7,193
1
vote
Asymptotic for a number theoretic sequence and its Dirichlet series' convergence
Your series is never absolutely convergent in any half-plane of the form $\mathrm{Re}\,s>\delta$ with $\delta<1$ and there is even no convergence in the case $\mathrm{Re}\,s=1$. To prove this, let us …
- 7,193
4
votes
How to get asymptotic expansion of the sum of modified Bessel function $\sum_{n=1}^\infty K_...
The term $\frac{\pi}{2s}$ comes from the pole of $\zeta(s)$. Let's use the fact that Mellin transform of $K_0(s)$ equals
$$
\int_0^{+\infty} K_0(s)s^{t-1}ds=2^{t-2}\Gamma^2(t/2).
$$
From this we get …
- 7,193