Partial fraction decomposition
The partial fraction decomposition or partial fraction is a standardized representation of rational functions. It is used in mathematics in order to facilitate the billing of such functions. In particular, it is used in the integration of the rational function for the application.
This is based on the fact that each rational function as a sum of a polynomial function and a fraction of the form
Can be displayed. The are the poles of the function.
If the poles assumed to be known, as is the determination of the numerator is the actual function of the partial fractions.
In real-valued functions, the poles and consequently the numbers may not necessarily be real, because the real numbers are not algebraically closed. It is calculating with complex numbers but avoid them because with every complex zero and the conjugate complex number is zero.
Instead of you and then uses a term, which is a real quadratic form and are also and real.
- 3.1 Simple poles
- 3.2 Double pole
- 3.3 Complex poles
- 4.1 Real-valued functions
- 4.2 Complex-valued functions
History
The partial fraction was developed from 1702 to work on the calculus of Gottfried Wilhelm Leibniz and Johann Bernoulli. Both scholars use this method for the integration of rational functions broken. At this time, the Fundamental Theorem of Algebra was not yet proven - but he was then already suspected - Leibniz asserted that there was no partial fraction expansion for the denominator. Johann Bernoulli did not follow this opinion. This example was discussed in subsequent years by various mathematicians and in 1720 published a number of works, which demonstrated the sample as faulty and the ( indefinite ) integral
Calculated correctly.
Method
The partial fraction expansion of a real rational function is determined in several steps:
- If the numerator is greater than or equal to the denominator, so, divide the numerator by the denominator. Obtained from the polynomial and possibly a rational residual function, so that:. , The process is finished.
- Otherwise, the counter of a smaller degree than the denominator. It then works only with the residual function on.
The latter two steps will now be explained in detail.
Approach
Is assumed here that there is in the mold, wherein the degree of less than the degree of the denominator polynomial, and all zeros of are known. If, as assumed above, the distinct zeros and their respective degrees known as the denominator polynomial can be brought to the following form:
Note that some can be the complex.
The approach is now structured as follows:
- For each simple real zero it contains a summand
- For each fold real root it contains summands
Since is real, belonging to each complex zero necessarily complex conjugate zeros. Be the quadratic polynomial with the zeros and, therefore.
- For each simple complex zero of the approach now includes a summand
- According to the approach for each fold complex zero ( and the associated also times, complex conjugate zero ) contains the terms
Each batch contains the same unknown coefficients
Determination of the constant
To determine the constants and is equated with the approach and multiplies this equation by the denominator.
On one side of the equation fails, only the numerator, on the other an expression with all the unknowns, which is a polynomial in and also can be arranged according as the powers of. A comparison of the coefficients of the left and right side then yields a linear system of equations from which the unknown constants can be calculated. Alternatively, you can connect up to any of various values for use in this equation, which results as the coefficients compared to existing equations of linear system of equations. It makes sense to the insertion of the previously calculated (real) zeros, which immediately provides each a coefficient value.
These two possibilities can be combined.
Examples
Simple poles
Consider the rational function
There are two simple poles and. The approach is therefore
Where and are unknown, yet to be determined constants. Multiplying with both sides of the equation, one obtains
Sort by you and members of the right-hand side without limbs, the result is
Coefficient comparison, the coefficient of is One: and the absolute zero element. From this it can be calculated: The required partial fraction expansion is therefore
Double pole
Consider the rational function
Using polynomial division and factorization of the denominator follows
The only, but double zero of the denominator is. approach:
Comparing the coefficients:
Solution:
So we get the partial fraction expansion
Complex poles
Consider the rational function
The denominator here is the real zero, the complex zero and its complex conjugate. The quadratic polynomial with the zeros and is
Approach:
Comparing the coefficients:
Solution:
Partial fractions:
The main theorem about partial fractions
Real-valued functions
Any rational function with distinct real poles of the order and the various up to conjugation complex poles of order has a uniquely determined representation
With a polynomial function and real constants, and. This is called the partial fraction expansion (abbreviated PBZ ) of.
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Complex-valued functions
Any rational function with the different poles of order has a uniquely determined representation
With a polynomial function and complex constants.
This theorem can be generalized for polynomials over any other skew field algebraically closed.
Applications
The partial fraction expansion is used among other things for integrating rational functions. Since the integrals of all the partial fractions are known, the integration is always possible if the conditions of poles of the function under consideration specify.
Furthermore, the partial fractions in the Laplacian, and the z- transformation is used. The transforms of the individual partial fractions can be looked up in tables. Thus one saves an analytical calculation, if the term to be transformed can be decomposed into corresponding summands.
Laurent series expansion
If for every pole a Laurent series expansion of the function is known, we obtain the partial fraction very simple sum of the main parts of this Laurent series. This path is related to the residue calculus.