Series development
A series development is a technique from mathematics that is particularly important in the sub-areas of analysis and function theory , but is also used in other mathematical disciplines as well as in physics and in other scientific and engineering fields.
In a series expansion, a mathematical function that cannot be represented directly with elementary operations ( addition , subtraction , multiplication and division ) is converted into an infinite sum of powers in one of its variables or of powers in another (usually elementary ) function.
In practice, this series can often be reduced to a finite number of terms. This results in an approximation of the exact function, which is the easier the fewer terms are taken, but the better the more are taken. The resulting inaccuracy (i.e. the size of the remainder of the link) can often be described using a formula.
In a generating function , the terms of an infinite sequence (e.g. that of Bernoulli's numbers ) appear as coefficients of the series expansion.
Examples
In mathematics, for example, the following series developments occur:
- Taylor series ( power series ) and, as a special case, Maclaurin series
- Laurent series : Generalization of the Taylor series, in which negative values of the exponents are also allowed.
- Puiseux series : Generalization of the Taylor series, in which fractional exponents are also allowed.
- Dirichlet series
- Fourier series : describes a periodic function as a superposition of sine and cosine functions. So z. B. Musical tones can be described as the superposition of a fundamental and several overtones .
- Legendre polynomial : in physics describes any field as a superposition of dipole , quadrupole , octupole fields etc. ( multipole expansion )
- Zernike polynomials : are in the optics used to image defects to calculate optical systems.
Other developments of such functions are the continued fraction developments .
Web links
- Eric W. Weisstein : Series Expansion . In: MathWorld (English).