Tangent half-angle substitution

In integral calculus, the tangent 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 unit circle parametrized by angle measure onto the real line. The general[1] transformation formula is:

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.[2] Leonhard Euler used it to evaluate the integral in his 1768 integral calculus textbook,[3] and Adrien-Marie Legendre described the general method in 1817.[4]

The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name.[5] It is known in Russia as the universal trigonometric substitution,[6] and also known by variant names such as half-tangent substitution or half-angle substitution. It is sometimes misattributed as the Weierstrass substitution.[7] Michael Spivak called it the "world's sneakiest substitution".[8]

  1. ^ Other trigonometric functions can be written in terms of sine and cosine.
  2. ^ Gunter, Edmund (1673) [1624]. The Works of Edmund Gunter. Francis Eglesfield. p. 73
  3. ^ 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.
    Also see Lobatto, Rehuel (1832). "19. Note sur l'intégration de la fonction z/(a + b cos z)". Crelle's Journal (in French). 9: 259–260.
  4. ^ Legendre, Adrien-Marie (1817). Exercices de calcul intégral [Exercises in integral calculus] (in French). Vol. 2. Courcier. p. 245–246.
  5. ^ Cite error: The named reference unnamed was invoked but never defined (see the help page).
  6. ^ Piskunov, Nikolai (1969). Differential and Integral Calculus. Mir. p. 379.
    Zaitsev, V. V.; Ryzhkov, V. V.; Skanavi, M. I. (1978). Elementary Mathematics: A Review Course. Ėlementarnai͡a matematika.English. Mir. p. 388.
  7. ^ Cite error: The named reference weierstrass was invoked but never defined (see the help page).
  8. ^ Spivak, Michael (1967). "Ch. 9, problems 9–10". Calculus. Benjamin. pp. 325–326.