Some Simple New Elementary Functions

In this post I introduce some new functions:

• $\mathrm{exph}(x)$
• $\mathrm{lnh}(x)$
• $\mathrm{cish}(x)$

I’m not sure whether to call these “hyperbolic” or “circular” versions of their sister functions.

The elementary functions include these (not a complete list)

• $\sin(x)$
• $\cos(x)$
• $\sin^{-1}(x)$
• $\cos^{-1}(x)$
• $\sinh(x) = \dfrac{e^x-e^{-x}}{2} = \dfrac{\exp(x)-\exp(-x)}{2}$
• $\cosh(x) = \dfrac{e^x+e^{-x}}{2} = \dfrac{\exp(x)+\exp(-x)}{2}$
• $\sinh^{-1}(y) = \ln\left[y+\sqrt{y^2+1}\right]$
• $\cosh^{-1}(y) = \ln\left[y+\sqrt{y^2-1}\right]$
• $\exp(x) = e^x = \sinh(x) +\cosh(x)$
• $\ln(x)$
• $\mathrm{cis}(x) = \cos(x) + i \sin(x)$

The following relationships and properties hold:

• $\sin(\cos^{-1}(x)) = \sqrt{1-x^2}$
• $\cos(\sin^{-1}(x)) = \sqrt{1-x^2}$ also
• $\sinh(x) = \sqrt{\cosh^2(x)-1}$
• $\cosh(x) = \sqrt{\sinh^2(x)+1}$
• $\sinh(\cosh^{-1}(x)) = \sqrt{x^2-1}$
• $\cosh(\sinh^{-1}(x)) = \sqrt{x^2+1}$
• $\sin(ix) = i \sinh(x)$
• $\cos(ix) = \cosh(x)$
• $\exp(0) = 1$
• $\ln(1) = 0$
• $\ln(ix) = \ln|x| + \dfrac{\pi i}{2}\mathrm{sgn}(x)$
• $\mathrm{cis}(x) = \exp(ix)$
• $\sinh^{-1}(y) = \cosh^{-1}\left(\sqrt{y^2+1}\right)$
• $\cosh^{-1}(y) = \sinh^{-1}\left(\sqrt{y^2-1}\right)$
• $\exp(x) = \sinh(x) + \sqrt{\sinh^2(x)+1}$
• $\exp(x) = \cosh(x) + \sqrt{\cosh^2(x)-1}$
• $\ln(x) = \sinh^{-1}\left(\dfrac{y^2-1}{2y}\right) = \cosh^{-1}\left(\dfrac{y^2+1}{2y}\right)$
• $\ln(x) = \tanh^{-1}\left(\dfrac{y^2-1}{y^2+1}\right)$
• $\dfrac{\mathrm{d}}{\mathrm{d}x} \sin(x) = \cos(x)$
• $\dfrac{\mathrm{d}}{\mathrm{d}x} \cos(x) = -\sin(x)$
• $\dfrac{\mathrm{d}^4}{\mathrm{d}x^4} \sin(x) = \sin(x)$
• $\dfrac{\mathrm{d}}{\mathrm{d}x} \sinh(x) = \cosh(x)$
• $\dfrac{\mathrm{d}}{\mathrm{d}x} \cosh(x) = \sinh(x)$
• $\dfrac{\mathrm{d}^2}{\mathrm{d}x^2} \sinh(x) = \sinh(x)$

Here are the new functions:

• $\mathrm{cish}(x) = \cosh(x) + i \sinh(x)$
• $\mathrm{exph}(x) = \sin(x) + \cos(x) = \sqrt{2}\sin\left(x+\dfrac{\pi}{4}\right)= \sqrt{2}\cos\left(x-\dfrac{\pi}{4}\right)$
• $\mathrm{lnh}(x) = \mathrm{exph}^{-1}(x)$
• $\mathrm{cish}(x) = \mathrm{exph}(ix)$

The following properties hold:

• $\mathrm{exph}(0) = 1$
• $\mathrm{lnh}(1) = 0$
• $\mathrm{lnh}(-1) = -\dfrac{\pi}{2}$
• $\sin(x) = \dfrac{\mathrm{exph}(x)-\mathrm{exph}(-x)}{2}$
• $\cos(x) = \dfrac{\mathrm{exph}(x)+\mathrm{exph}(-x)}{2}$
• $\mathrm{lnh}(x) = \sin^{-1}\left(\dfrac{x}{\sqrt{2}}\right) - \dfrac{\pi}{4}$
• $\mathrm{Lnh}(x) = \sin^{-1}\left(\dfrac{x}{\sqrt{2}}\right) - \dfrac{\pi}{4} + 2n\pi$ for $n \in \mathbb{Z}$
• $\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{exph}(x) = \mathrm{exph}(-x)$
• $\dfrac{\mathrm{d}^2}{\mathrm{d}x^2} \mathrm{exph}(x) = -\mathrm{exph}(x)$
• $\dfrac{\mathrm{d}^4}{\mathrm{d}x^4} \mathrm{exph}(x) = \mathrm{exph}(x)$

I don’t think these functions will turn out to be of any importance.

However, they do seem to help complete the set.

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