PlanetMath: Weierstrass substitution formulas

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Weierstrass substitution formulas

(Definition)

The Weierstrass substitution formulas for $-\pi < x < \pi$ are:

$\begin{displaymath}\begin{array}{rl} \sin x & =\displaystyle \frac{2t}{1+t^2} \\... ...\ & \ dx & =\displaystyle \frac{2}{1+t^2} \, dt \end{array}\end{displaymath}$

They can be obtained in the following manner:

Make the Weierstrass substitution $\displaystyle t=\tan \left( \frac{x}{2} \right)$ . (This substitution is also known as the universal trigonometric substitution.) Then we have

$\begin{displaymath}\begin{array}{rl} \displaystyle \cos \left( \frac{x}{2} \righ... ... \ & \ & = \displaystyle \frac{1}{\sqrt{1+t^2}} \end{array}\end{displaymath}$

and

$\begin{displaymath}\begin{array}{rl} \displaystyle \sin \left( \frac{x}{2} \righ... ...\ & \ & = \displaystyle \frac{t}{\sqrt{1+t^2}}. \end{array}\end{displaymath}$

Note that these are just the “formulas involving radicals” as designated in the entry goniometric formulas; however, due to the restriction on , the $\pm$ 's are unnecessary.

Using the above formulas along with the double angle formulas, we obtain

$\begin{displaymath}\begin{array}{rl} \sin x & =\displaystyle 2\sin\left( \frac{x... ...1+t^2}} \ & \ & =\displaystyle \frac{2t}{1+t^2} \end{array}\end{displaymath}$

and

$\begin{displaymath}\begin{array}{rl} \cos x & =\displaystyle \cos^2\left( \frac{... ...^2} \ & \ & =\displaystyle \frac{1-t^2}{1+t^2}. \end{array}\end{displaymath}$

Finally, since $\displaystyle t=\tan\left( \frac{x}{2} \right)$ , solving for yields that $x=2\arctan t$ . Thus, $\displaystyle dx=\frac{2}{1+t^2} \, dt$ .

The Weierstrass substitution formulas are most useful for integrating rational functions of sine and cosine.

"Weierstrass substitution formulas" is owned by Wkbj79.

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Other names:

Weierstraß substitution formulas

Also defines:

Weierstrass substitution, Weierstaß substitution, universal trigonometric substitution

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Cross-references: double angle formulas, goniometric formulas
There are 2 references to this entry.

This is version 9 of Weierstrass substitution formulas, born on 2007-05-14, modified 2007-05-30.
Object id is 9383, canonical name is WeierstrassSubstitutionFormulas.
Accessed 2236 times total.

Classification:

AMS MSC:	33B10 (Special functions :: Elementary classical functions :: Exponential and trigonometric functions)
	26A36 (Real functions :: Functions of one variable :: Antidifferentiation)

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