Difference between revisions of "Hankel H (2) in terms of csc and Bessel J"
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(Created page with "==Theorem== The following formula holds: $$H_{\nu}^{(2)}(z)=i\csc(\nu \pi) \left[ J_{-\nu}(z)-e^{\nu \pi i}J_{\nu}(z) \right],$$ where $H_{\nu}^{(2)}$ denotes the Hankel H (...") |
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==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Hankel H (2)|next=}}: 9.1.4 | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Hankel H (2)|next=Relationship between Bessel J sub n and Bessel J sub -n}}: 9.1.4 |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 04:50, 11 June 2016
Theorem
The following formula holds: $$H_{\nu}^{(2)}(z)=i\csc(\nu \pi) \left[ J_{-\nu}(z)-e^{\nu \pi i}J_{\nu}(z) \right],$$ where $H_{\nu}^{(2)}$ denotes the Hankel function of the second kind, $i$ denotes the imaginary number, $\csc$ denotes the cosecant, $J_{-\nu}$ denotes the Bessel function of the first kind, and $e^{\nu \pi i}$ denotes the exponential.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 9.1.4
