PlanetMath: Weierstrass substitution formulas

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (21)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

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.

(view preamble | get metadata)

Other names:

Weierstraß substitution formulas

Also defines:

Weierstrass substitution, Weierstaß substitution, universal trigonometric substitution

Log in to rate this entry.
(view current ratings)

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 3254 times total.

Classification:

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

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact