算術幾何平均
M ( a , b ) = π 2 / ∫ 0 π / 2 d θ ( a + b 2 ) 2 cos 2  θ + a b sin 2  θ = π 2 / ∫ 0 π / 2 d θ ( a + b 2 ) 2 − ( a − b 2 ) 2 sin 2  θ = π 2 / ∫ 0 π / 2 d θ ( a + b 2 ) 1 − ( a − b a + b ) 2 sin 2  θ {\displaystyle {\begin{aligned}M(a,b)&={\frac {\pi }{2}}{\bigg /}\int _{0}^{\pi /2}{\frac {d\theta }{\sqrt {\left({\frac {a+b}{2}}\right)^{2}\cos ^{2}\theta +ab\sin ^{2}\theta }}}\\&={\frac {\pi }{2}}{\bigg /}\int _{0}^{\pi /2}{\frac {d\theta }{\sqrt {\left({\frac {a+b}{2}}\right)^{2}-\left({\frac {a-b}{2}}\right)^{2}\sin ^{2}\theta }}}\\&={\frac {\pi }{2}}{\bigg /}\int _{0}^{\pi /2}{\frac {d\theta }{\left({\frac {a+b}{2}}\right){\sqrt {1-\left({\frac {a-b}{a+b}}\right)^{2}\sin ^{2}\theta }}}}\\\end{aligned}}}
a , b {\displaystyle a,\ b} が複素数である場合は、積分路 t = b x 2 − a 2 a b x {\displaystyle t={\frac {bx^{2}-a}{2{\sqrt {ab}}\;x}}} と実軸との間に(留数をもつ)極がないことを確かめなければならない。 u = R ( b / a ) {\displaystyle u=\Re \left(b/a\right)} , v = I ( b / a ) {\displaystyle v=\Im \left(b/a\right)} とすれば、
t a + b 2 a b = ( b / a ) x 2 − 1 ( 1 + b / a ) x = ( u + i v ) x 2 − 1 ( 1 + u + i v ) x = ( u x 2 − 1 + i v x 2 ) ( 1 + u − i v ) ( ( 1 + u ) 2 + v 2 ) x = ( u + u 2 + v 2 ) x 2 − ( 1 + u ) + i v x 2 + i v ( 1 + 2 u + u 2 + v 2 ) x {\displaystyle {\begin{aligned}{\frac {t}{\frac {a+b}{2{\sqrt {ab}}}}}&={\frac {(b/a)x^{2}-1}{(1+b/a)x}}={\frac {(u+iv)x^{2}-1}{(1+u+iv)x}}\\&={\frac {(ux^{2}-1+ivx^{2})(1+u-iv)}{\left((1+u)^{2}+v^{2}\right)x}}\\&={\frac {(u+u^{2}+v^{2})x^{2}-(1+u)+ivx^{2}+iv}{(1+2u+u^{2}+v^{2})x}}\\\end{aligned}}}