Weierstrass Substitution


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Proof Technique

The 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:

Sine

$\sin \theta = \dfrac {2 u} {1 + u^2}$


Cosine

$\cos \theta = \dfrac {1 - u^2} {1 + u^2}$


Derivative

$\dfrac {\d \theta} {\d u} = \dfrac 2 {1 + u^2}$


The above results 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 $u = \tan \dfrac \theta 2$.


Also known as

The technique of 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 $1$.
  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Indefinite Integrals: Important Transformations: $14.58$
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): half-angle formulae: 1.
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): half-angle formulae: 1.
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 16$: Indefinite Integrals: Important Transformations: $16.58.$
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