Tangent half-angle substitution: Difference between revisions
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In [[integral calculus]], the ''' |
In [[integral calculus]], the '''Weierstrass substitution''',<ref name=":0" /><ref>Weisstein, Eric W. "[https://mathworld.wolfram.com/WeierstrassSubstitution.html Weierstrass Substitution]." From ''MathWorld''--A Wolfram Web Resource. Accessed April 1, 2020.</ref><ref>{{cite journal |last1=Jeffrey |first1=David J. |last2=Rich |first2=Albert D. |year=1994 |title=The evaluation of trigonometric integrals avoiding spurious discontinuities |url=https://dl.acm.org/doi/pdf/10.1145/174603.174409 |journal=Transactions on Mathematical Software |volume=20 |issue=1 |pages=124-135}}</ref>, also known as the '''tangent half-angle substitution''', '''universal trigonometric substitution''',<ref>{{cite book |last=Piskunov |first=Nikolai |author-link= Nikolai Piskunov |title=Differential and Integral Calculus |publisher=Mir |year=1969}} [https://archive.org/details/n.-piskunov-differential-and-integral-calculus-mir-1969/page/379/ p. 379]</ref> the '''half-tangent substitution''', or the '''half-angle substitution''', is a [[Integration by substitution|change of variables]] used for evaluating [[antiderivative|integrals]], which converts a [[rational function]] of [[trigonometric functions]] of <math display=inline>x</math> into an ordinary rational function of <math display=inline>t</math> by setting <math display=inline>t = \tan \tfrac x2</math>. This is the one-dimensional [[stereographic projection]] of the circle onto the line. The general transformation formula is:<ref>Other trigonometric functions can be written in terms of sine and cosine.</ref> |
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|last1=Jeffrey |first1=David J. |last2=Rich |first2=Albert D. |year=1994 |title=The evaluation of trigonometric integrals avoiding spurious discontinuities |journal=Transactions on Mathematical Software |volume=20 |issue=1 |pages=124-135 |url=https://dl.acm.org/doi/pdf/10.1145/174603.174409}}</ref> '''half-tangent substitution''', or '''half-angle substitution''', is a [[Integration by substitution|change of variables]] used for evaluating [[antiderivative|integrals]], which converts a [[rational function]] of [[trigonometric functions]] of <math display=inline>x</math> into an ordinary rational function of <math display=inline>t</math> by setting <math display=inline>t = \tan \tfrac x2</math>. This is the one-dimensional [[stereographic projection]] of the circle onto the line. The general transformation formula is:<ref>Other trigonometric functions can be written in terms of sine and cosine.</ref> |
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<math display=block>\int f(\sin x, \cos x)\, dx =\int f{\left(\frac{2t}{1+t^2},\frac{1-t^2}{1+t^2}\right)} \frac{2\,dt}{1+t^2}.</math> |
<math display=block>\int f(\sin x, \cos x)\, dx =\int f{\left(\frac{2t}{1+t^2},\frac{1-t^2}{1+t^2}\right)} \frac{2\,dt}{1+t^2}.</math> |
Revision as of 16:54, 21 September 2022
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In integral calculus, the Weierstrass substitution,[1][2][3], also known as the tangent half-angle substitution, universal trigonometric substitution,[4] the half-tangent substitution, or the half-angle substitution, is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of into an ordinary rational function of by setting . This is the one-dimensional stereographic projection of the circle onto the line. The general transformation formula is:[5]
History
The tangent of half an angle is important in spherical trigonometry and was sometimes known in the 17th century as the half tangent or semi-tangent.[6] Leonhard Euler used it to solve the integral in his 1768 integral calculus textbook,[7] and Adrien-Marie Legendre described the general method in 1817.[8] James Stewart named it after Karl Weierstrass when discussing the substitution in his popular 1987 calculus textbook.[1] Michael Spivak wrote that this method was the "sneakiest substitution" in the world.[9]
The substitution
Introducing a new variable sines and cosines can be expressed as rational functions of , and can be expressed as the product of and a rational function of , as follows:
Derivation of the formulas
Using the double-angle formulas, introducing denominators equal to one thanks to Pythagorean theorem, and then dividing numerators and denominators by one gets
Finally, since , the differentiation rules imply
and thus
Examples
First example: the cosecant integral
We can confirm the above result using a standard method of evaluating the cosecant integral by multiplying the numerator and denominator by and performing the substitution .
These two answers are the same because
The secant integral may be evaluated in a similar manner.
Second example: a definite integral
In the first line, one does not simply substitute for both limits of integration. The singularity (in this case, a vertical asymptote) of at must be taken into account. Alternatively, first evaluate the indefinite integral then apply the boundary values. By symmetry, which is the same as the previous answer.
Third example: both sine and cosine
if
Geometry
As x varies, the point (cos x, sin x) winds repeatedly around the unit circle centered at (0, 0). The point
goes only once around the circle as t goes from −∞ to +∞, and never reaches the point (−1, 0), which is approached as a limit as t approaches ±∞. As t goes from −∞ to −1, the point determined by t goes through the part of the circle in the third quadrant, from (−1, 0) to (0, −1). As t goes from −1 to 0, the point follows the part of the circle in the fourth quadrant from (0, −1) to (1, 0). As t goes from 0 to 1, the point follows the part of the circle in the first quadrant from (1, 0) to (0, 1). Finally, as t goes from 1 to +∞, the point follows the part of the circle in the second quadrant from (0, 1) to (−1, 0).
Here is another geometric point of view. Draw the unit circle, and let P be the point (−1, 0). A line through P (except the vertical line) is determined by its slope. Furthermore, each of the lines (except the vertical line) intersects the unit circle in exactly two points, one of which is P. This determines a function from points on the unit circle to slopes. The trigonometric functions determine a function from angles to points on the unit circle, and by combining these two functions we have a function from angles to slopes.
Gallery
Hyperbolic functions
As with other properties shared between the trigonometric functions and the hyperbolic functions, it is possible to use hyperbolic identities to construct a similar form of the substitution, :
Geometrically, this change of variables is a one-dimensional analog of the Poincaré disk projection.
See also
- Rational curve
- Stereographic projection
- Tangent half-angle formula
- Trigonometric substitution
- Euler substitution
Further reading
- Courant, Richard (1937) [1934]. "1.4.6. Integration of Some Other Classes of Functions §1–3". Differential and Integral Calculus. Vol. 1. Blackie & Son. pp. 234–237.
- Edwards, Joseph (1921). "§1.6.193". A Treatise on the Integral Calculus. Vol. 1. Macmillan. pp. 187–188.
- Hardy, Godfrey Harold (1905). "VI. Transcendental functions". The integration of functions of a single variable. Cambridge. pp. 42–51. Second edition 1916, pp. 52–62
- Hermite, Charles (1873). "Intégration des fonctions transcendentes" [Integration of transcendental functions]. Cours d'analyse de l'école polytechnique (in French). Vol. 1. Gauthier-Villars. pp. 320–380.
- Weierstrass, Karl (1915). "8. Bestimmung des Integrals ...". Mathematische Werke von Karl Weierstrass [Mathematical Works of Karl Weierstrass] (in German). Vol. 6. Mayer & Müller. pp. 89–99.
Notes and references
- ^ a b Stewart, James (1987). Calculus. Brooks/Cole. p. 431.
The German mathematician Karl Weierstrauss (1815–1897) noticed that the substitution t = tan(x/2) will convert any rational function of sin x and cos x into an ordinary rational function.
- ^ Weisstein, Eric W. "Weierstrass Substitution." From MathWorld--A Wolfram Web Resource. Accessed April 1, 2020.
- ^ Jeffrey, David J.; Rich, Albert D. (1994). "The evaluation of trigonometric integrals avoiding spurious discontinuities". Transactions on Mathematical Software. 20 (1): 124–135.
- ^ Piskunov, Nikolai (1969). Differential and Integral Calculus. Mir. p. 379
- ^ Other trigonometric functions can be written in terms of sine and cosine.
- ^ Gunter, Edmund (1673) [1624]. The Works of Edmund Gunter. Francis Eglesfield. p. 73
- ^ Euler, Leonhard (1768). "§1.1.5.261 Problema 29" (PDF). Institutiones calculi integralis [Foundations of Integral Calculus] (in Latin). Vol. I. Impensis Academiae Imperialis Scientiarum. pp. 148–150. E342, Translation by Ian Bruce.
- ^ Legendre, Adrien-Marie (1817). Exercices de calcul intégral [Excercises in integral calculus] (in French). Vol. 2. Courcier. p. 245–246.
- ^ Spivak, Michael (1967). "Ch. 9, problems 9–10". Calculus. Benjamin. pp. 325–326.