Series expansion

A series expansion is a technique from mathematics, which is particularly in subdivisions analysis and function theory of meaning, but is also used in other mathematical disciplines. In a series expansion is a mathematical function that can not be represented directly with elementary operations (addition, subtraction, multiplication and division), transferred to an infinite sum of powers in one of its variables or of powers in a different (usually elementary ) function.

This so-called number can be reduced to a finite number of terms, in practice, often, which is an approximation to the exact function is formed that is easier the less limbs were taken, but is the better, the more were taken. Can often be the thereby resulting inaccuracy ( ie the size of the residual limb ) describe formulaic.

A generating function for the members of a infinite sequence (e.g., the Bernoulli numbers ) appear as coefficients of the series expansion.

Examples

In mathematics, for example, enter the following series expansions on:

• Taylor series ( power series ) and as a special case of Maclaurin series
• Laurent series: Generalization of the Taylor series, wherein the negative values ​​of the exponents are allowed.
• Puiseuxreihe: Generalization of the Taylor series, and in the broken exponents are allowed.
• Dirichlet series
• The Fourier series describing a periodic function as a superposition of sine and cosine functions. Thus, for example Musical sounds will be described as a superposition of a fundamental and a number of overtones.
• Legendre polynomial: describes in physics any field as a superposition of dipole, quadrupole, octupole field, etc. ( multipole )
• Zernike polynomials are used in optics to calculate aberrations of optical systems.

Other developments such functions are the continued fractions.