Entire function

In the theory of functions is an entire function is a function that is holomorphic in the whole complex plane (ie analytical) is. Typical examples of entire functions are polynomials or exponential function, as well as sums, products, and links them, such as the trigonometric functions and the hyperbolic functions.

Properties

Every entire function can be represented as an everywhere convergent power series to any center. Neither the logarithm or the square root function are all about.

An entire function, an isolated singularity, in particular even an essential singularity in the complex point at infinity (and only there) own.

An important property of entire functions is the set of Liouville: Is an entire function is limited, it must be constant, so you can quite elegantly prove the fundamental theorem of algebra. The small set of Picard is a considerable tightening of the set of Liouville: a non-constant entire function takes on all values ​​of the complex plane, except possibly one. The latter is illustrated, for example, except for the exponential function that never takes the value 0.

Other examples

  • The inverse 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 integer
  • The Struve functions for integer
  • The greatest common divisor with respect to a natural number in the generalized form
Complex analysis Analytic function Liouville's theorem (complex analysis) Reinhold Remmert Wolfram Research
360529
de