Fourier analysis notation - Sh and Ch

Question

I reading something dealing with Fourier analysis and don't know what "Sh" and "Ch" indicate. Thanks!

Answer 1

The hyperbolic functions are the "real" counterparts of the ordinary trigonometric ones.

$$\text{ch}(x)=\cosh(x)=\frac{e^x+e^{-x}}2\leftrightarrow \cos(x)=\frac{e^{ix}+e^{-ix}}2,$$

$$\text{sh}(x)=\sinh(x)=\frac{e^x-e^{-x}}2\leftrightarrow \sin(x)=\frac{e^{ix}-e^{-ix}}{2i}.$$

They are odd and even linear combinations of the exponential, so they easily appear with the latter.

Their name stems form the relation $$c^2-s^2=1$$ that corresponds to an hyperbola, to be compared to

$$c^2+s^2=1$$ for the circular functions.