Weierstrass substitution formulas

The Weierstrass substitution formulas for -π<x<π are:


They can be obtained in the following manner:

Make the Weierstrass substitution t=tan(x2). (This substitution is also known as the universal trigonometric substitution.) Then we have




Note that these are just the “formulas involving radicalsMathworldPlanetmath (http://planetmath.org/Radical6)” as designated in the entry goniometric formulasPlanetmathPlanetmath; however, due to the restriction on x, the ±’s are unnecessary.

Using the above formulas along with the double angle formulas, we obtain




Finally, since t=tan(x2), solving for x yields that x=2arctant. Thus, dx=21+t2dt.

The Weierstrass substitution formulas are most useful for integrating rational functions of sine and cosine (http://planetmath.org/IntegrationOfRationalFunctionOfSineAndCosine).

Title Weierstrass substitution formulas
Canonical name WeierstrassSubstitutionFormulas
Date of creation 2013-03-22 17:05:25
Last modified on 2013-03-22 17:05:25
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 12
Author Wkbj79 (1863)
Entry type Definition
Classification msc 26A36
Classification msc 33B10
Synonym Weierstraß substitution formulas
Related topic GoniometricFormulae
Related topic IntegrationOfRationalFunctionOfSineAndCosine
Related topic PolynomialAnalogonForFermatsLastTheorem
Defines Weierstrass substitution
Defines Weierstaß substitution
Defines universal trigonometric substitution