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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