Elementary Functions

  • Elementary, my dear Watson, elementary!…It was easier to know it than to explain why I know it.If you were asked to prove that two and two made four, you might find some difficulty, and yet you are quitesure of the fact…In solving a problem of this sort, the grand thing is to be able to reason backward. That is a very useful accomplishment, and a very easy one, but people do not practise it much. In the everyday affairs of life it is more useful to reason forward, and so the other comes to be neglected. There are fifty who can reason synthetically for one who can reason analytically…How often have I said to you that when you haveeliminated the impossible, whatever remains, however improbable, must be the truth?…I never guess.It is a shocking habit, destructive to the logical faculty… You know my methods. Apply them. (Sherlock Holmes)

In general, so called elementary functions, such as the exponential function, are defined as solutions of certain (basic elementary) differential equations, can be computed, decimal by decimal, by time stepping with smaller and smaller time step. There are lots of elementary functions, since there are lots of possible more or less elementary differential equations, many of them named after famous mathematicians, including

  • Bessel functions
  • Hyperbolic functions
  • Chebyshev polynomials
  • Legendre polynomials
  • Jacobi functions
  • Hermite functions
  • Laguerre functions
  • Hankel functions
  • Elliptic functions
  • Gamma function
  • Beta function
  • Riemann zeta function

For example, the Bessel functions J_\alpha (x) are solutions x(t) to the differential equation

  • t^2\ddot x+t\dot x+(t^2-\alpha^2)x=0

where \alpha is a constant, which arises for in problems with cylindrical or spherical symmetry.

Bessel functions J_\alpha (x).

The time step required to reach a certain precision or number of decimals, can vary from one differential equation and elementary function to the other. Below we shall study this problem, that is the dependence on the solution of differential equation on the time step used to compute it.

We can only compute a finite number of decimals of \exp(t), as many as our computational resources allows, but we can never list all of the decimals of \exp(t), expect for specific values such as \exp(0)=1.

We can think of \exp(t) as unique number, with a possibly never repeating decimal expansion, but we should be aware of the fact that this a kind illusion because we can never pin down exactly what \exp(t) is, except in the implicit form of saying that it is the function u(t) with the property to solve \dot u(t) =u(t) for t>0 with u(0)=1.

In old times, the values of elementary functions were listed in printed mathematical tables obtained by time-stepping the corresponding differential equations. In a computer, the values of elementary functins are not stored in tables but are recomputed every time a valu is requested.

                                              Hermite crab function

To Think About

  • How are elementary functions computed by your computer?


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