integration of (1/x-x)^alpha
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Hi,
here is my code
function y = f(r, alpha)
F = @(x) (1./x-x).^alpha;
y = integral(F,0,r);
end
Then if i type
f(1/2,0.2)
i get
ans =
0.707008459146274
but the problems start when alpha gets closer to 1, for example:
f(1/2, 0.8)
Warning: Infinite or Not-a-Number value
encountered.
> In funfun\private\integralCalc>iterateScalarValued at 349
In funfun\private\integralCalc>vadapt at 132
In funfun\private\integralCalc at 75
In integral at 88
In f at 3
ans =
Inf
How can I compute this integral? Thank you for reading
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Accepted Answer
Roger Stafford
on 11 Jun 2014
Yes, you are right, Camille. If alpha < 1, your function is integrable, but the infinity in its integrand at x = 0 is apparently proving to be too difficult for matlab's 'integral' function to evaluate it properly.
However, if you make the following simple change of variable, you should encounter no such difficulty. To simplify notation use 'a' instead of 'alpha'. Define variable z as:
z = x^(1-a)
Then we have
dz = (1-a)/x^a*dx
and
(1/x-x)^a*dx = (1-x^2)^a/x^a*dx = (1-z^(2/(1-a)))^a/(1-a)*dz
Hence you should evaluate the integral
F = @(z) (1-z^(2/(1-a)))^a/(1-a);
y = integral(F,0,r^(1-a));
Matlab should have no trouble with this integral provided a < 1, since it has no singularities in its integrand.
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More Answers (1)
Sean de Wolski
on 11 Jun 2014
1./1-1 yields a 0 and then 0 raised to any negative power is inf. What do you expect at 1?
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