Hyperbolic Functions

from University of Tokyo 1S1 Mathematical Science Foundation: Differential and Integral Calculus Hyperbolic Functions

• sinh, cosh, tanh
  • The inverse functions are called sech, csch, coth.
• $\sinh x:=\frac{e^x-e^{-x}}{2}$, $\cosh x:=\frac{e^x+e^{-x}}{2}$
  • $\tanh x:=\frac{\sinh x}{\cosh x}$, so $\tanh x=\frac{e^x-e^{-x}}{e^x+e^{-x}}$
  • It feels familiar (blu3mo)
  • If I mix in i instead of x, it becomes the trigonometric functions of a circle, right? (nishio)
    • Euler’s formula
    • That’s right (takker)
      • i governs rotation
      • So when you put i in hyperbolic functions, they rotate, and when you remove i from trigonometric functions, they stop rotating.
    • I see (blu3mo)
      • However, I didn’t know this, so the source of the familiarity might be different.
      • Just my imagination?
      • Maybe it appears in quadratic curves in Math III? (takker)
      • It might also appear in the integration of rational functions.
        • I think so (blu3mo)
        • Shall we try differentiating trigonometric functions expressed in exponential notation? (nishio)
  • Why is this written in a trigonometric-like form? (blu3mo)
  • The symmetry/relationship is not yet apparent (blu3mo)
  • The properties are similar
    • ${\cos }^{2}x+{\sin }^{2}x=1$ / ${\cosh }^{2}x-{\sinh }^{2}x=1$
      • They are inverses
      • In a sense, they are very similar (nishio)
      • If all the other formulas have the opposite signs, they feel similar, but it doesn’t seem so (blu3mo)
      • The opposites are between circular functions (trigonometric functions) and hyperbolic functions. By comparing their definitions, it becomes very, very clear (takker)
        • I have a feeling that checking the definitions from the properties will make sense (blu3mo)
  • The shapes are not that similar, right?
    • They are not approximating each other
  • Instead of the circle $x^2+y^2=1$, they are defined using the hyperbola $x^2-y^2=1$?
  • It’s interesting around here (takker)
    • Let’s throw in some advanced topics
    • The operational rules of $\mathrm{Re}(z):=\frac{z+z^{\ast }}{2}$, $\mathrm{Im}(z):=\frac{z-z^{\ast }}{2i}$
    • The operational rules of $\frac{x+x^{-1}}{2}$, $\frac{x-x^{-1}}{2}$
    • It helps to understand which properties are due to $e$, which are due to $i$, and which are due to $\frac{a\pm b}{2}$

• Trigonometric functions and hyperbolic functions are connected when extended to complex numbers.

  • The relationship between trigonometric functions and hyperbolic functions is more clearly understood by considering the power series obtained by Taylor expansion in the complex number range.

  • They said they will explain in detail later
  • Oh no (blu3mo)
    • I might be able to self-study since I’ve looked into this before
  • 7 Connection with complex numbers
    • I see!!! (blu3mo)
      • Does it take the same form as Representation of trigonometric functions using Euler’s formula?
    • $\cosh z=\frac{e^z+e^{-z}}{2}$ $\sinh z=\frac{e^z-e^{-z}}{2}$
      • cos replaces ix with z
      • If you cut the three-dimensional graph of cosh with a plane parallel to the imaginary axis and perpendicular to the real axis, you get the graph of cos(x)?
        • That seems to be the case (blu3mo)
        • 
        • Not all values of cosh(z) return real numbers (blu3mo)
          • If you want to plot all the values of cosh(z), you need a four-dimensional plot
    • I think I understand the feelings of hyperbolic functions (blu3mo)(blu3mo)(blu3mo)
    • If we write the laws of hyperbolic functions based on trigonometric functions, it might be easier to understand
      • If we write the laws of hyperbolic functions and trigonometric functions using exp, it might be even better