Weierstrass Substitution

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Proof Technique

The Weierstrass Substitution is an application of Integration by Substitution.


The substitution is:

$u \leftrightarrow \tan \dfrac \theta 2$

for $-\pi < \theta < \pi$, $u \in \R$.


It yields:

\(\displaystyle \sin \theta\) \(=\) \(\displaystyle \frac {2 u} {1 + u^2}\)
\(\displaystyle \cos \theta\) \(=\) \(\displaystyle \frac {1 - u^2} {1 + u^2}\)
\(\displaystyle \frac {\d \theta} {\d u}\) \(=\) \(\displaystyle \frac 2 {1 + u^2}\)


This can be stated:

$\displaystyle \int \map F {\sin \theta, \cos \theta} \rd \theta = 2 \int \map F {\frac {2 u} {1 + u^2}, \frac {1 - u^2} {1 + u^2} } \frac {d u} {1 + u^2}$

where $u = \tan \dfrac \theta 2$.


Proof

Let $u = \tan \dfrac \theta 2$ for $-\pi < \theta < \pi$.

From Shape of Tangent Function, this substitution is valid for all real $u$.

Then:

\(\displaystyle u\) \(=\) \(\displaystyle \tan \dfrac \theta 2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \theta\) \(=\) \(\displaystyle 2 \tan^{-1} u\) Definition of Inverse Tangent
\(\displaystyle \leadsto \ \ \) \(\displaystyle \dfrac {\d \theta} {\d u}\) \(=\) \(\displaystyle \dfrac 2 {1 + u^2}\) Derivative of Arctangent Function and Derivative of Constant Multiple
\(\displaystyle \sin \theta\) \(=\) \(\displaystyle \dfrac {2 u} {1 + u^2}\) Tangent Half-Angle Substitution for Sine
\(\displaystyle \cos \theta\) \(=\) \(\displaystyle \dfrac {1 - u^2} {1 + u^2}\) Tangent Half-Angle Substitution for Cosine


The result follows from Integration by Substitution.

$\blacksquare$


Also known as

This technique is also known as tangent half-angle subsitution.


Also see


Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.


Sources

This article incorporates material from Weierstrass substitution formulas on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.