Introduction

• What do we teach our students:
  • semesters 1 - 3
    • only analytic computations
    • analysis, linear algebra, ordinary differential equations
    • applications (mechanics, electrical engineering, vibration theory)
  • semesters 4 - 7
    • real-world problems are solved numerically
    • numerical mathematics, FEM, modeling and simulation
  • never
    • how to extend analytical methods
    • special functions
• Which functions do we use:
  • elementary arithmetic → rational functions
  • powers → algebraic functions
  • equations with powers → exponential function + logarithm
  • geometry → sin, cos, tan + inverses arcsin etc
  • that's it (sinh etc. is nothing new)
• Definition of new functions:
  • in mathematics: lots and lots of strange stuff (Weierstrass function!)
  • focus here: driven by applications and general usefulness
  • construction methods used here
    • inverse functions
    • antiderivatives
    • definite integrals
    • solutions of ODEs
  • lots of other possibilities, e.g.
    • power series
    • integral transforms (Fourier, Laplace, ...)
  • here only functions on ℝ
  • no systematics (e.g. hypergeometric series)