In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a series expansion is a method for calculating a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division). The resulting so-called series

Series (mathematics)

A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

often can be limited to a finite number of terms, thus yielding an approximation

Approximation

An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...

 of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.e., the partial sum of the omitted terms) can be described by an equation.

There are several kinds of series expansions, such as:

• Taylor series
  Taylor series
  In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
  
  : A power series based on a function’s derivative
  Derivative
  In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
  
  s at a single point.
• Maclaurin series: A special case of a Taylor series, centred at zero.
• Laurent series
  Laurent series
  In mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where...
  
  : An extension of the Taylor series, allowing negative exponent values.
• Dirichlet series: Used in number theory
  Number theory
  Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
  
  .
• Fourier series
  Fourier series
  In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
  
  : Describes periodical functions as a series of sine and cosine functions. In acoustics
  Acoustics
  Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...
  
  , e.g., the fundamental tone
  Fundamental frequency
  The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
  
   and the overtone
  Overtone
  An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...
  
  s together form an example of a Fourier series.
• Legendre polynomials: Used in physics
  Physics
  Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
  
   to describe an arbitrary electrical field as a superposition
  Superposition
  Superposition can refer to:* The superposition principle in physics, mathematics, and engineering, describes the overlapping of waves. Particular applications include :** Quantum superposition, in quantum physics** Superposition theorem, in electronics....
  
   of a dipole
  Dipole
  In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...
  
   field, a quadrupole
  Quadrupole
  A quadrupole or quadrapole is one of a sequence of configurations of—for example—electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity.-Mathematical...
  
   field, an octupole
  Multipole expansion
  A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original...
  
   field, etc.
• Zernike polynomials
  Zernike polynomials
  In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after Frits Zernike, they play an important role in beam optics.-Definitions:There are even and odd Zernike polynomials...
  
  : Used in optics
  Optics
  Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
  
   to calculate aberration
  Aberration
  An aberration is something that deviates from the normal way.Aberration may refer to:In optics and physics:*Optical aberration, an imperfection in image formation by an optical system...
  
  s of optical systems. Each term in the series describes a particular type of aberration.

For more details, refer to the articles mentioned.