John2012-04-27 10:13:19

Hi,

In a problem I’ve been working on recently, I have the following integral

int_0^{a} int _0^{b} { frac{y}{ x^2 + y^2 – 2xyc } } dx dy

where 0<=c<1 There is a singularity as x,y approach 0. I need to compute this integral fairly accurately and quickly. I've spent a lot of time trying to isolate the singularity in a convenient form, but have so far been unsuccessful. Does anyone have any ideas about how to go about computing this integral. I haven't been able to find a closed form expression, but there might be one. Thanks in advance. John

Spellucci2012-04-27 10:13:36

In article <3281hrF3ivmvhU1@individual.net>,

John

int _0^b { frac{y}{ x^2 + y^2 – 2xyc } } dx =

= 1/sqrt(1-c^2)*(arctan( (b-y*c)/(y*sqrt(1-c^2)) ) + arctan(c/sqrt(1-c^2) )

but integrating this one from 0 to a seems not to be possible in closed form.

maybe this helps

(I hope I got it right)

peter

Rusin2012-04-27 10:13:58

Polar coordinates leads to a rational function of sines and cosines;

the half-angle trick makes it the integral of a rational function;

use partial fractions.

dave

Paul abbott2012-05-04 00:40:34

Mathematica gives

( a ArcTan[(b – a c)/(a Sqrt[1 – c^2])] +

b c ArcTan[(a – b c)/(b Sqrt[1 – c^2])] +

(a + b c) ArcSin[c])/Sqrt[1 – c^2] +

b/2 Log[(a^2 – 2 b c a + b^2)/b^2]
For example, with a -> 1, b -> 2, and c -> 1/2, one obtains

4 Pi/(3 Sqrt[3]) – Log[4/3]

which to 20 decimals is 2.1307170…

Doing the integral numerically,

NIntegrate[y/(x^2 + y^2 – 2 c y x) /. c -> 1/2, {x, 0, 2}, {y, 0, 1}]

2.1307170…

Cheers,

Paul

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