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

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