Difference between revisions of "Relationship between sinh and sin"
From specialfunctionswiki
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==Proof== | ==Proof== | ||
| + | By definition, | ||
| + | $$\sinh(z) = \dfrac{e^{z}-e^{-z}}{2},$$ | ||
| + | and so by the definition of $\sin$ and the fact that $-i=\dfrac{1}{i}$, we see | ||
| + | $$-i\sinh(iz)=\dfrac{e^{iz}-e^{-iz}}{2i},$$ | ||
| + | as was to be shown. █ | ||
==References== | ==References== | ||
| Line 10: | Line 15: | ||
[[Category:Theorem]] | [[Category:Theorem]] | ||
| − | [[Category: | + | [[Category:Proven]] |
Revision as of 01:04, 25 June 2016
Theorem
The following formula holds: $$\sinh(z)=-i\sin(iz),$$ where $\sinh$ is the hyperbolic sine and $\sin$ is the sine.
Proof
By definition, $$\sinh(z) = \dfrac{e^{z}-e^{-z}}{2},$$ and so by the definition of $\sin$ and the fact that $-i=\dfrac{1}{i}$, we see $$-i\sinh(iz)=\dfrac{e^{iz}-e^{-iz}}{2i},$$ as was to be shown. █
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.5.7
