Tangent half-angle substitution (English Wikipedia)

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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. E 342, Translation by Ian Bruce.
Also see Lobatto, Rehuel (1832). "19. Note sur l'intégration de la fonction $\partial z / (a + b cos z)$ ". Crelle's Journal (in French). 9: 259–260.

archive.org

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. E 342, Translation by Ian Bruce.
Also see Lobatto, Rehuel (1832). "19. Note sur l'intégration de la fonction $\partial z / (a + b cos z)$ ". Crelle's Journal (in French). 9: 259–260.
Legendre, Adrien-Marie (1817). Exercices de calcul intégral [Exercises in integral calculus] (in French). Vol. 2. Courcier. p. 245–246.
For example, in chronological order,
- Hermite, Charles (1873). Cours d'analyse de l'école polytechnique (in French). Vol. 1. Gauthier-Villars. p. 320.
- 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.
- In 1966 William Eberlein attributed this substitution to Karl Weierstrass (1815–1897):
  Eberlein, William Frederick (1966). "The Circular Function(s)". Mathematics Magazine. 39 (4): 197–201. doi:10.1080/0025570X.1966.11975715. JSTOR 2688079. (Equations (3) [ $x=\cos \theta$ ], (4) [ $y=\sin \theta$ ], (5) [ $t=\tan {\tfrac {\theta }{2}}$ ] are, of course, the familiar half-angle substitutions introduced by Weierstrass to integrate rational functions of sine, cosine.)
  Two decades later, James Stewart mentioned Weierstrass when discussing the substitution in his popular calculus textbook, first published in 1987:
  Stewart, James (1987). "§7.5 Rationalizing substitutions". Calculus. Brooks/Cole. p. 431. ISBN 9780534066901. The German mathematician Karl Weierstrass (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.
  
  Later authors, citing Stewart, have sometimes referred to this as the Weierstrass substitution, for instance:
  - Jeffrey, David J.; Rich, Albert D. (1994). "The evaluation of trigonometric integrals avoiding spurious discontinuities". Transactions on Mathematical Software. 20 (1): 124–135. doi:10.1145/174603.174409. S2CID 13891212.
  - Spivak, Michael (1967). "Ch. 9, problems 9–10". Calculus. Benjamin. pp. 325–326.
  doi.org
  - In 1966 William Eberlein attributed this substitution to Karl Weierstrass (1815–1897):
    Eberlein, William Frederick (1966). "The Circular Function(s)". Mathematics Magazine. 39 (4): 197–201. doi:10.1080/0025570X.1966.11975715. JSTOR 2688079. (Equations (3) [ $x=\cos \theta$ ], (4) [ $y=\sin \theta$ ], (5) [ $t=\tan {\tfrac {\theta }{2}}$ ] are, of course, the familiar half-angle substitutions introduced by Weierstrass to integrate rational functions of sine, cosine.)
    Two decades later, James Stewart mentioned Weierstrass when discussing the substitution in his popular calculus textbook, first published in 1987:
    Stewart, James (1987). "§7.5 Rationalizing substitutions". Calculus. Brooks/Cole. p. 431. ISBN 9780534066901. The German mathematician Karl Weierstrass (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.
    
    Later authors, citing Stewart, have sometimes referred to this as the Weierstrass substitution, for instance:
    
    Jeffrey, David J.; Rich, Albert D. (1994). "The evaluation of trigonometric integrals avoiding spurious discontinuities". Transactions on Mathematical Software. 20 (1): 124–135. doi:10.1145/174603.174409. S2CID 13891212.
    hathitrust.org
    catalog.hathitrust.org
    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.
    jstor.org
    In 1966 William Eberlein attributed this substitution to Karl Weierstrass (1815–1897):
    Eberlein, William Frederick (1966). "The Circular Function(s)". Mathematics Magazine. 39 (4): 197–201. doi:10.1080/0025570X.1966.11975715. JSTOR 2688079. (Equations (3) [ $x=\cos \theta$ ], (4) [ $y=\sin \theta$ ], (5) [ $t=\tan {\tfrac {\theta }{2}}$ ] are, of course, the familiar half-angle substitutions introduced by Weierstrass to integrate rational functions of sine, cosine.)
    Two decades later, James Stewart mentioned Weierstrass when discussing the substitution in his popular calculus textbook, first published in 1987:
    Stewart, James (1987). "§7.5 Rationalizing substitutions". Calculus. Brooks/Cole. p. 431. ISBN 9780534066901. The German mathematician Karl Weierstrass (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.
    
    Later authors, citing Stewart, have sometimes referred to this as the Weierstrass substitution, for instance:
    
    Jeffrey, David J.; Rich, Albert D. (1994). "The evaluation of trigonometric integrals avoiding spurious discontinuities". Transactions on Mathematical Software. 20 (1): 124–135. doi:10.1145/174603.174409. S2CID 13891212.
    maa.org
    eulerarchive.maa.org
    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. E 342, Translation by Ian Bruce.
    Also see Lobatto, Rehuel (1832). "19. Note sur l'intégration de la fonction $\partial z / (a + b cos z)$ ". Crelle's Journal (in French). 9: 259–260.
    pacific.edu
    scholarlycommons.pacific.edu
    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. E 342, Translation by Ian Bruce.
    Also see Lobatto, Rehuel (1832). "19. Note sur l'intégration de la fonction $\partial z / (a + b cos z)$ ". Crelle's Journal (in French). 9: 259–260.
    semanticscholar.org
    api.semanticscholar.org
    In 1966 William Eberlein attributed this substitution to Karl Weierstrass (1815–1897):
    Eberlein, William Frederick (1966). "The Circular Function(s)". Mathematics Magazine. 39 (4): 197–201. doi:10.1080/0025570X.1966.11975715. JSTOR 2688079. (Equations (3) [ $x=\cos \theta$ ], (4) [ $y=\sin \theta$ ], (5) [ $t=\tan {\tfrac {\theta }{2}}$ ] are, of course, the familiar half-angle substitutions introduced by Weierstrass to integrate rational functions of sine, cosine.)
    Two decades later, James Stewart mentioned Weierstrass when discussing the substitution in his popular calculus textbook, first published in 1987:
    Stewart, James (1987). "§7.5 Rationalizing substitutions". Calculus. Brooks/Cole. p. 431. ISBN 9780534066901. The German mathematician Karl Weierstrass (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.
    
    Later authors, citing Stewart, have sometimes referred to this as the Weierstrass substitution, for instance:
    
    Jeffrey, David J.; Rich, Albert D. (1994). "The evaluation of trigonometric integrals avoiding spurious discontinuities". Transactions on Mathematical Software. 20 (1): 124–135. doi:10.1145/174603.174409. S2CID 13891212.
    
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Tangent half-angle substitution (English Wikipedia)

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doi.org

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