Euler's Addition Theorem

Author

Eric W. Weisstein
January 25, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/EulersAdditionTheorem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/EulersAdditionTheorem.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Check

N[With[{a=Random[]/10,b=Random[]/5,c=Random[]/4,m=Random[Complex]},{EllipticF[ArcSin[a],m]+EllipticF[ArcSin[b],m]-EllipticF[ArcSin[(b*Sqrt[(1-a^2)*(1-m*a^2)]+a*Sqrt[(1-b^2)*(1-m*b^2)])/(1-m*a^2*b^2)],m]}]]

With[{k=Random[Complex],a=Random[Complex],b=Random[Complex]},c=(b*Sqrt[(1-a^2)*(1-k^2*a^2)]+a*Sqrt[(1-b^2)*(1-k^2*b^2)])/(1-k^2*a^2*b^2);
{Integrate[1/Sqrt[(1-t^2)*(1-k^2*t^2)],{t,0,c}],Integrate[1/Sqrt[(1-t^2)*(1-k^2*t^2)],{t,0,a}]+Integrate[1/Sqrt[(1-t^2)*(1-k^2*t^2)],{t,0,b}]}]