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Re: Different results for same integration


"ashesh" <ashesh.cb at gmail.com> wrote:
> Hi all,
>
> I am trying to do the following two integrations, which are basically
> the same, but with a change of variable. I am getting different results
> from both of them. Hope some one can point out the mistake I am making.
>
> a = 19.0; b = 4.0; t = 5.0;
>
> Integrate[(a + b)/Sqrt[(a^2 - x^2)*(b^2 - x^2)], {x, b, b + I*t}]
>
> Integrate[(1 + b/a)/Sqrt[(1 - y^2)*(1 - (b^2*y^2)/a^2)], {y, 1, (b +
> I*t)/b}]
>
> where y = (x/b)
>
> The first integration gives: -1.23787 + 1.44831 I
>
> while the second one gives: 6.17818 - 5.4757 I
>
> The upper limits of the integrations are complex (b + i t) and ((b + i
> t)/b) respectively.
>
> The result from the first integration is correct and I have verified it
> analytically.

I suggest that you avoid using inexact numbers for a, b and t. The integral
is unstable at its lower limit. (In fact, it mught be luck that your first
integral gave the correct numerical answer.)

For your second integration, I recommend

In[9]:= a = 19; b = 4; t = 5;
NIntegrate[(1 + b/a)/Sqrt[(1 - y^2)*(1 - (b^2*y^2)/a^2)], {y, 1, (b + I*t)/b}]

Out[9]= -1.23787 + 1.44831 I

David


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