Solving int_0^infty (1+(y_1^2+y_2^2+...+y_n^2)/v dy_1dy_2... dy_n

tun1ju2k1ki 2022-09-06 Answered
Solving 0 ( 1 + y 1 2 + y 2 2 + + y n 2 ) ν ) d y 1 d y 2 d y n
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Answers (1)

ordonansexa
Answered 2022-09-07 Author has 7 answers
As you have noticed, the integral can be transformed into
0 d n y 1 ( 1 + y T y ) α .
Going to the spherical coordinates leads to
1 2 n d Ω n 0 d r r n 1 ( 1 + r 2 ) α ,
where the factor of 1 / 2 n is due to integrating over this fraction of the whole space and d Ω n is the volume of S n 1 , which is equal to
d Ω n = 2 π n / 2 Γ ( n 2 ) ,
as can be shown for example by evaluating the integral d n y e y T y both in Cartesian and spherical coordinates and comparing the two results.
To evaluate the simple integral over r, make the change of variables x = ( 1 + r 2 ) 1 , which leads to
0 d r r n 1 ( 1 + r 2 ) α = 1 2 0 1 x α n / 2 1 ( 1 x ) n / 2 1 d x = Γ ( α n / 2 ) Γ ( n / 2 ) 2 Γ ( α ) ,
where the expression for the Beta function was used in the last step. Putting the pieces together, you arrive at
0 d n y 1 ( 1 + y T y ) α = ( π 4 ) n / 2 Γ ( α n / 2 ) Γ ( α ) .
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