Weierstrass substitution

The Weierstraß substitution (also known as the half-angle method) is a method from the mathematical branch of analysis . It is a variant of integration by substitution that can be applied to certain integrands with trigonometric functions . The method is named after the mathematician Karl Weierstrass , who developed it.

Description of the substitution

Let be two real numbers and a rational function . To be an integral of the form ${\displaystyle a<b}$ ${\displaystyle R}$

${\displaystyle \int _{a}^{b}R(\sin(x),\cos(x))\,dx}$

can calculate the substitution

${\displaystyle \tan \left({\frac {x}{2}}\right)=t}$

to be applied for. The substitutions then result for the functions sine and cosine ${\displaystyle |x|<\pi }$

${\displaystyle {\begin{aligned}\sin x&={\frac {2t}{1+t^{2}}}\\\cos x&={\frac {1-t^{2}}{1+t^{2}}}\end{aligned}}}$

and for the differential applies

${\displaystyle dx={\frac {2\,dt}{1+t^{2}}}}$.

Since the functions tangent , cotangent , secant and cosecant can be written as fractions with sine and cosine, the Weierstrass substitution can be applied to these trigonometric functions. The substitutions are ${\displaystyle \tan }$ ${\displaystyle \cot }$ ${\displaystyle \sec }$ ${\displaystyle \csc }$

${\displaystyle {\begin{aligned}\tan x&={\frac {2t}{1-t^{2}}}\\\cot x&={\frac {1-t^{2}}{2t}}\\\csc x&={\frac {1+t^{2}}{2t}}\\\sec x&={\frac {1+t^{2}}{1-t^{2}}}\,.\end{aligned}}}$

Alternatively, an integral of the above form can also be solved in a function-theoretic way. The real interval is transformed into a complex area and then the residual theorem is applied.

example

The general substitution is suitable for eliminating the trigonometric functions when calculating the integral, as the following example shows.

${\displaystyle \int {\frac {2}{3+\cos(x)}}\,dx=\int {\frac {2}{(3+{\frac {1-t^{2}}{1+t^{2}}})}}\cdot {\frac {2}{(1+t^{2})}}\,dt=\int {\frac {4}{3+3t^{2}+1-t^{2}}}\,dt=\int {\frac {2}{2+t^{2}}}\,dt}$

This integral can now be calculated with a further integration by substitution .

Derivation

In this section the substitution formulas for sine and cosine are derived. With the addition theorems we get:

${\displaystyle \sin(x)=2\sin \left({\frac {x}{2}}\right)\cos \left({\frac {x}{2}}\right)=2t\cdot \cos ^{2}\left({\frac {x}{2}}\right)}$

${\displaystyle t={\frac {\sin({\frac {x}{2}})}{\cos({\frac {x}{2}})}}\Longrightarrow \cos ^{2}\left({\frac {x}{2}}\right)={\frac {1-\cos ^{2}\left({\frac {x}{2}}\right)}{t^{2}}}\Longrightarrow \cos ^{2}\left({\frac {x}{2}}\right)={\frac {1}{1+t^{2}}}}$.

Together you have the illustration above for . The representation for is obtained as follows: ${\displaystyle \sin(x)}$ ${\displaystyle \cos(x)}$

${\displaystyle \cos(x)={\sqrt {1-\sin ^{2}(x)}}={\sqrt {1-{\frac {4t^{2}}{(1+t^{2})^{2}}}}}={\frac {1-t^{2}}{1+t^{2}}}}$for , ${\displaystyle x\in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}$

${\displaystyle \cos(x)=-{\sqrt {1-\sin ^{2}(x)}}=-{\sqrt {1-{\frac {4t^{2}}{(1+t^{2})^{2}}}}}={\frac {1-t^{2}}{1+t^{2}}}}$for . ${\displaystyle x\in \left[{\frac {\pi }{2}},{\frac {3\pi }{2}}\right]}$

The derivation of to results with: ${\displaystyle x}$ ${\displaystyle t}$

${\displaystyle t=\tan \left({\frac {x}{2}}\right)\iff x=2\cdot \arctan(t)\Rightarrow {\frac {dx}{dt}}={\frac {2}{1+t^{2}}}}$.

Web links

• Eric W. Weisstein : Weierstrass Substitution . In: MathWorld (English).

Individual evidence

1. ^ Howard Anton, Irl Bivens, Stephen Davis: Calculus . 9th edition. John Wiley & Sons, Inc., 2009, ISBN 978-0-470-18345-8 , pp. 526-528 .