Difference between revisions of "Orthogonality of Bateman F on R"
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(Created page with "==Theorem== The following formula holds: $$\displaystyle\int_{-\infty}^{\infty} F_m(it)F_n(it) \mathrm{sech}^2\left(\dfrac{\pi t}{2} \right)\mathrm{d}t = \dfrac{4(-1)^n}{\pi (...") |
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Latest revision as of 11:51, 10 October 2019
Theorem
The following formula holds: $$\displaystyle\int_{-\infty}^{\infty} F_m(it)F_n(it) \mathrm{sech}^2\left(\dfrac{\pi t}{2} \right)\mathrm{d}t = \dfrac{4(-1)^n}{\pi (2n+1)} \delta_{mn},$$ where $F_n$ denotes the Bateman F, $i$ denotes the imaginary number, $\mathrm{sech}$ denotes sech, $\pi$ denotes pi, and $\delta$ denotes Kronecker delta.
