# 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:

\(\displaystyle \sin \theta\) | \(=\) | \(\displaystyle \frac {2 u} {1 + u^2}\) | |||||||||||

\(\displaystyle \cos \theta\) | \(=\) | \(\displaystyle \frac {1 - u^2} {1 + u^2}\) | |||||||||||

\(\displaystyle \frac {\d \theta} {\d u}\) | \(=\) | \(\displaystyle \frac 2 {1 + u^2}\) |

This can be stated:

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

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:

\(\displaystyle u\) | \(=\) | \(\displaystyle \tan \dfrac \theta 2\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \theta\) | \(=\) | \(\displaystyle 2 \tan^{-1} u\) | Definition of Inverse Tangent | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \dfrac {\d \theta} {\d u}\) | \(=\) | \(\displaystyle \dfrac 2 {1 + u^2}\) | Derivative of Arctangent Function and Derivative of Constant Multiple | |||||||||

\(\displaystyle \sin \theta\) | \(=\) | \(\displaystyle \dfrac {2 u} {1 + u^2}\) | Tangent Half-Angle Substitution for Sine | ||||||||||

\(\displaystyle \cos \theta\) | \(=\) | \(\displaystyle \dfrac {1 - u^2} {1 + u^2}\) | 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 subsitution**.

## Also see

## Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 14$: Important Transformations: $14.58$

*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. http://mathworld.wolfram.com/WeierstrassSubstitution.html