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.
  • ,
    • , so
    • 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
      • /
        • 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 , they are defined using the hyperbola ?
    • It’s interesting around here (takker)
      • Let’s throw in some advanced topics
      • The operational rules of ,
      • The operational rules of ,
      • It helps to understand which properties are due to , which are due to , and which are due to
  • 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)
        • 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)
          • image
          • 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