Weierstrass Substitution/Derivative
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Proof Technique
Let:
- $u \leftrightarrow \tan \dfrac \theta 2$
for $-\pi < \theta < \pi$, $u \in \R$.
Then:
- $\dfrac {\d \theta} {\d u} = \dfrac 2 {1 + u^2}$
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 |
$\blacksquare$
Also known as
The technique of Weierstrass Substitution is also known as Tangent Half-Angle Substitution.
Some sources call these results the Tangent-of-Half-Angle Formulae.
Other sources refer to them 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 $1$.
- This article incorporates material from Weierstrass substitution formulas on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.