Euler substitutions

Euler permutations are permutations reducing integrals of the form $\int R(x,{\sqrt {ax^{2}+bx+c}})dx$ ${\ displaystyle \ int R (x, {\ sqrt {ax ^ {2} + bx + c}}) dx}$ $\ int R (x, \ sqrt {ax ^ 2 + bx + c}) dx$ where $R(x,{\sqrt {ax^{2}+bx+c}})$ ${\ displaystyle R (x, {\ sqrt {ax ^ {2} + bx + c}})}$ $R (x, \ sqrt {ax ^ 2 + bx + c})$ - a rational function, to the integrals of rational functions. Proposed by L. Euler in 1768 ^[1] ^[2] .

Content

1 Substitution
- 1.1 First substitution
- 1.2 Second Substitution
- 1.3 Third Substitution
2 Interesting Facts
3 notes
4 References

Substitutions

First Substitution

Used when $a>0$ ${\ displaystyle a> 0}$ . Replacing:
${\sqrt {ax^{2}+bx+c}}=\pm t\pm {\sqrt {a}}x$ ${\ displaystyle {\ sqrt {ax ^ {2} + bx + c}} = \ pm t \ pm {\ sqrt {a}} x}$

Second Substitution

Used when $c>0$ ${\ displaystyle c> 0}$ . Replacing:
${\sqrt {ax^{2}+bx+c}}=\pm xt\pm {\sqrt {c}}$ ${\ displaystyle {\ sqrt {ax ^ {2} + bx + c}} = \ pm xt \ pm {\ sqrt {c}}}$

Third Substitution

Used when the root expression has two valid roots. Replacing:
${\sqrt {ax^{2}+bx+c}}=\pm t(x-\lambda )$ ${\ displaystyle {\ sqrt {ax ^ {2} + bx + c}} = \ pm t (x- \ lambda)}$ where $\lambda$ ${\ displaystyle \ lambda}$ - one of the roots ^[1] .

Interesting Facts

According to the memoirs of a student of Landau A.I. Akhiezer , he was extremely negative about the use of these substitutions:

<...> he [Landau] suggested that I calculate <...> the integral of a rational fraction. <...> I calculated, without using Euler’s standard permutations, and this saved me, because, as I understood later, Landau did not tolerate them and thought that every time I had to use some kind of artificial technique, which, in fact, I did.
- Memoirs of L. D. Landau ^[3]

Notes

↑ ¹ ² Euler Substitutions // Big Soviet Encyclopedia / Ch. ed. A.M. Prokhorov . - 3rd ed. - M .: Soviet Encyclopedia , 1978. - T. 29: Chagan - Aix-les-Bains. - S. 575. - 632,000 copies.
↑ Leonhardo Eulero. Institutionum calculi integralis . - Petropolis , 1768. - Vol. 1. - P. 57-61.
↑ Memoirs of L.D. Landau / Otv. ed. Acad. I.M.Khalatnikov. - Anthology. - M .: Nauka , 1988 .-- S. 49 .-- 354 p. - 23,100 copies. - ISBN 5-02-000091-4 .

Links

[BSE-1] ¹ ² Euler Substitutions // Big Soviet Encyclopedia / Ch. ed. A.M. Prokhorov . - 3rd ed. - M .: Soviet Encyclopedia , 1978. - T. 29: Chagan - Aix-les-Bains. - S. 575. - 632,000 copies.

[2] Leonhardo Eulero. Institutionum calculi integralis . - Petropolis , 1768. - Vol. 1. - P. 57-61.

[3] Memoirs of L.D. Landau / Otv. ed. Acad. I.M.Khalatnikov. - Anthology. - M .: Nauka , 1988 .-- S. 49 .-- 354 p. - 23,100 copies. - ISBN 5-02-000091-4 .