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
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration: Useful substitutions: : Example $1$.
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Important Transformations: $14.58$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: half-angle formulae: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: half-angle formulae: 1.
- This article incorporates material from Weierstrass substitution formulas on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
- Weisstein, Eric W. "Weierstrass Substitution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WeierstrassSubstitution.html