Power series

Under a power series is understood in analysis an infinite series of the form

With

• Any sequence of real or complex numbers
• The expansion point of the power series.

Power series play an important role in the theory of functions and often allow a meaningful continuation of real functions in the complex plane. In particular, the question arises, for which real or complex numbers converges as a power series. This question leads to the concept of the radius of convergence.

Radius of convergence

As the radius of convergence of a power series around the point of development the largest number is defined by the power series converges for all for which. The open ball with radius is called circle of convergence. The radius of convergence is therefore the radius of the circle of convergence. If the series converges for all, then we say that the radius of convergence is infinite. Converges only for so is the radius of convergence 0, this is sometimes nowhere called convergent.

For power series can be the radius of convergence of the formula of Cauchy -Hadamard calculate. The following applies:

In this context, is defined and

In many cases, the radius of convergence can be easily calculated in the following manner with a power series with non-zero coefficients. It is namely

Provided that this limit exists.

Examples

Each polynomial can be regarded as power series with infinite radius of convergence, where all coefficients with the exception of a finite number are 0. Other important examples are also Taylor series and Maclaurin series. Functions that can be represented by power series are also called analytic functions. Here is an example of the power series representation of some known functions:

• Exponential function: for all, that is the radius of convergence is infinite.

• Sine:

• Cosine:

• Logarithm function:

• Root function: for, that is, the radius of convergence is 1 and the series converges both for and for.

Properties

Power series are normally convergent within its circle of convergence. It follows directly that each defined by a power series function is continuous. Furthermore, it follows that on compact subsets of the circle of convergence is present uniform convergence. This justifies the term by term differentiation and integration of power series, showing that power series are infinitely differentiable. Furthermore, within the circle of convergence remains the absolute convergence. About the behavior of a power series on the edge of the circle of convergence, no general statement can be made, but in some cases it allows the Abelian limit theorem, to make a statement.

The power series representation of a function at a point of development is uniquely determined ( identity theorem for power series ). In particular, for a given development point the Taylor expansion of the only existing power series representation.

Operations with power series

Addition and scalar multiplication

Are and two power series

With the radius of convergence and is a real or complex number, then also, and again power series with radius of convergence at least and it is

Multiplication

The product of two power series with radius of convergence is also a power series with radius of convergence is at least. Since inside the circle of convergence absolute convergence is present, then applies after the Cauchy product formula

The result is referred to as convolution or convolution of two sequences and.

Differentiation and integration

A power series is differentiable in the interior of its circle of convergence and the derivative is obtained by termwise differentiation.

Here is infinitely differentiable and we have:

Analogously, one obtains an antiderivative by term by term integration of a power series.

In both cases, the radius of convergence corresponding to the original number.

Representation of functions as power series

Often one is interested in a given function in a power series representation, in particular, to answer the question of whether the function is analytic. There are several strategies to determine a power series representation. The most common is by means of the Taylor series. Here, however, often the problem occurs that requires a closed form for the derivatives, which is often difficult to determine. For fractional rational functions but there are some lighter strategies. As an example, the function

Be considered.

By factoring the denominator and then applying the geometric series formula we obtain the following representation of the function as a product of infinite series

Both series are power series around the point of development and therefore can be multiplied in the manner mentioned above. The same result also provides the Cauchy product formula

With

And

It follows by applying the geometric sum

As a closed form for the sequence of coefficients of the power series. Thus, the power series representation of the function to the development of point 0 is given by

Often the path via the geometric series is cumbersome and error-prone. Therefore, the following approach offers: It is believed that a power series representation of the function of unknown coefficient sequence exists

After Durchmultiplizieren the denominator and an index shift results in the identity

However, since two power series if and are equal if their coefficient sequences are identical, it follows by comparing coefficients and the recurrence equation, from which by induction follows the above closed-form representation.

The procedure by comparing coefficients also has the advantage that other development points as possible. Consider as an example the development point. First, the rational function must be represented as a polynomial in:

As above, it is assumed now that a formal power series around the point of development there with an unknown coefficient sequence and multiplied by the denominator by:

.

Again results by comparing coefficients and as a recurrence equation for the coefficients

.

Applying to the given polynomial function first and then the partial fraction expansion of, we obtain the representation

Substituting the geometric series results

The first three terms of the sequence of the sequence of coefficients are all zero, and thus determines the representation given here with the top match.

Generalizations

Power series can not only define, but are also generalized. Thus, for example, the Matrixexponential and the matrix logarithm generalizations of power series on the space of square matrices. Looking at negative exponent, it is called a Laurent series. Allowing the exponent to take fractional values ​​, it is a Puiseauxreihe.

Formal power series are used for example as generating functions in combinatorics and probability theory (such as the probability- generating functions ). In algebra, formal power series over commutative rings generally be investigated.