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