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MathGroup Archive 2007

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Re: Different results for same integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg72973] Re: [mg72964] Different results for same integration
  • From: leigh pascoe <leigh at cephb.fr>
  • Date: Sat, 27 Jan 2007 05:41:38 -0500 (EST)
  • References: <200701261242.HAA10459@smc.vnet.net>

ashesh 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.
>
> Looking forward for any help in resolving the problem.
>
> Ashesh
>
>
>
>   
It helps to do these symbolically first

a = 19; b = 4; t = 5;
Integrate[(a + b)/Sqrt[(a^2 - x^2)*(b^2 - x^2)], {x, b, b + I*t}]

Out: \!\(â?«\_4\%\(4 + 5\ \[ImaginaryI]\)\(23\/\@\(\((16 - x\^2)\)\ \((361 -
x\^2)\)\)\) \[DifferentialD]x\)

x = b*y;
Integrate[(1 + b/a)/Sqrt[(1 - y^2)*(1 - (b^2*y^2)/a^2)], {y, 1, (b + 
I*t)/b}]

Out: \!\(â?«\_1\%\(1 + \(5\ \[ImaginaryI]\)\/4\)\(23\/\(19\ \@\(\((1 - 
y\^2)\)\ \((1 \
- \(16\ y\^2\)\/361)\)\)\)\) \[DifferentialD]y\)

I can't get Ma to do this for me, but if you take the second expression 
and put y=4x and dy=4dx, you get the integral of

23/â??(16-x^2)(361-16x^2)

whereas the first expression evaluates to the integral of

23/â??(16-x^2)(361-x^2)

So these don't seem to be equivalent expressions.

Leigh


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