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:

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


This can be stated:

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

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:

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

Some sources refer to these results merely as the half-angle formulas or half-angle formulae.


Also see


Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.


Sources