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Euler's addition theorem for elliptic integrals states that, with ,

where

For real variables with and , use positive square roots and require the sum of the two integrals to lie between and , where is the complete elliptic integral of the first kind. The formula follows from the addition identity for the Jacobi elliptic function (DLMF 22.8.1).

A form of Euler's differential equation is the separable differential equation

where is a polynomial of degree at most four (Euler 1761, §30). On real intervals where is positive, take positive square roots. If , separation gives for a constant . The addition theorem above gives this sum as when , , , and .

See also

Abel's Addition Theorem, Elliptic Integral of the First Kind, Jacobi Elliptic Functions

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References

Euler, L. "De integratione aequationis differentialis ." Novi Comment. Acad. Sci. Imp. Petropol. 6, 37-57, 1761. https://scholarlycommons.pacific.edu/euler-works/251/.NIST Digital Library of Mathematical Functions. "Addition Theorems." https://dlmf.nist.gov/22.8.

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Euler's Addition Theorem

Cite this as:

Weisstein, Eric W. "Euler's Addition Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EulersAdditionTheorem.html

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