Whole function

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In function theory , a whole function is a function that is holomorphic (i.e. analytical ) in the entire complex number plane . Typical examples of whole functions are polynomials or the exponential function as well as sums, products and combinations thereof, such as the trigonometric functions and the hyperbolic functions .


Every whole function can be represented as an everywhere converging power series around any center. Neither the logarithm nor the root function are whole.

A whole function can have an isolated singularity , in particular even an essential singularity in the complex point at infinity (and only there).

An important property of whole functions is Liouville's theorem : Bounded whole functions are constant . This is a very elegant way to prove the fundamental theorem of algebra . The small set of Picard is a substantial tightening of the set of Liouville: A non-constant entire function accepts all the values of the complex plane, except for possibly a. The latter exception illustrates, for example, the exponential function , which does not take the value 0.

Further examples

The Airy function (here the real part) is a whole function.
  • the reciprocal of the gamma function
  • the error function
  • the integral sine
  • the Airy functions and
  • the Fresnel integrals and
  • the Riemann Xi function
  • the Bessel functions of the first kind for whole numbers
  • the Struve functions for integers
  • the greatest common divisor with respect to a natural number in the generalized form
    • ( Ramanujan sum )


  • Klaus Jänich : Function Theory . An introduction . 6th edition. Springer-Verlag, Berlin / Heidelberg 2004
  • Reinhold Remmert : Analysis I . 3. Edition. Springer-Verlag, Berlin Heidelberg 1992
  • Eberhard Freitag , Rolf Busam: Function Theory 1 . 3. Edition. Springer-Verlag, Berlin / Heidelberg 2000

Web links

Individual evidence

  1. ^ Wolfgang Schramm: The Fourier transform of functions of the greatest common divisor . In: Integers - The Electronic Journal of Combinatorial Number Theory , 8, 2008, A50 ( abstract )