Integral of a complicated rational function
$begingroup$
I have a tough integral that I would like your advice on.
Let $varepsilon>0$ be real, $i>0$ an integer, and $gamma = gamma(varepsilon)$ some (known) complicated function of $varepsilon$. Essentially, I need to compute the following integral:
$$int_{a(varepsilon)}^{b(varepsilon)}frac{x^i}{(1+x^2)^2 big((x+gamma)^2+(1-gamma^2) big)^{i-1}}, dx$$
for some equally complicated (and also known) bounding functions $a,b$.
My gut reaction is to compute a brute force partial fraction decomposition and work term-by-term, but this quickly becomes messy. Is there something "smart" I can do instead? I suspect there may be a way to handle it more elegantly with residue techniques, but I am not fluent enough to know how, and I couldn't find a similar example in my go-to complex analysis texts.
Thanks!
EDIT: Also suppose none of the poles lie between $a$ and $b$; that should be reasonable based on my problem.
integration
asked Jan 11 at 3:54
MathIsArtMathIsArt
1278
$endgroup$
add a comment |
$begingroup$
I have a tough integral that I would like your advice on.
Let $varepsilon>0$ be real, $i>0$ an integer, and $gamma = gamma(varepsilon)$ some (known) complicated function of $varepsilon$. Essentially, I need to compute the following integral:
$$int_{a(varepsilon)}^{b(varepsilon)}frac{x^i}{(1+x^2)^2 big((x+gamma)^2+(1-gamma^2) big)^{i-1}}, dx$$
for some equally complicated (and also known) bounding functions $a,b$.
My gut reaction is to compute a brute force partial fraction decomposition and work term-by-term, but this quickly becomes messy. Is there something "smart" I can do instead? I suspect there may be a way to handle it more elegantly with residue techniques, but I am not fluent enough to know how, and I couldn't find a similar example in my go-to complex analysis texts.
Thanks!
EDIT: Also suppose none of the poles lie between $a$ and $b$; that should be reasonable based on my problem.
integration
asked Jan 11 at 3:54
MathIsArtMathIsArt
1278
$endgroup$
$begingroup$
I don't expect there would be a particular nice general form. If we denote the integrand by $f_i(x)$, then we have $f_{i + 1}(x) = frac{x}{x^2 + 2 gamma x + 1} f_i(x)$, which suggests trying to set up a recursion relation in $i$ and relating the partial fractions decompositions of $f_i, f_{i + 1}$.
$endgroup$
– Travis
Jan 13 at 23:18
$begingroup$
Thanks for your suggestion, I appreciate it. I will try this.
$endgroup$
– MathIsArt
Jan 14 at 1:20
add a comment |
$begingroup$
I have a tough integral that I would like your advice on.
Let $varepsilon>0$ be real, $i>0$ an integer, and $gamma = gamma(varepsilon)$ some (known) complicated function of $varepsilon$. Essentially, I need to compute the following integral:
$$int_{a(varepsilon)}^{b(varepsilon)}frac{x^i}{(1+x^2)^2 big((x+gamma)^2+(1-gamma^2) big)^{i-1}}, dx$$
for some equally complicated (and also known) bounding functions $a,b$.
My gut reaction is to compute a brute force partial fraction decomposition and work term-by-term, but this quickly becomes messy. Is there something "smart" I can do instead? I suspect there may be a way to handle it more elegantly with residue techniques, but I am not fluent enough to know how, and I couldn't find a similar example in my go-to complex analysis texts.
Thanks!
EDIT: Also suppose none of the poles lie between $a$ and $b$; that should be reasonable based on my problem.
integration
asked Jan 11 at 3:54
MathIsArtMathIsArt
1278
$endgroup$
I have a tough integral that I would like your advice on.
Let $varepsilon>0$ be real, $i>0$ an integer, and $gamma = gamma(varepsilon)$ some (known) complicated function of $varepsilon$. Essentially, I need to compute the following integral:
$$int_{a(varepsilon)}^{b(varepsilon)}frac{x^i}{(1+x^2)^2 big((x+gamma)^2+(1-gamma^2) big)^{i-1}}, dx$$
for some equally complicated (and also known) bounding functions $a,b$.
My gut reaction is to compute a brute force partial fraction decomposition and work term-by-term, but this quickly becomes messy. Is there something "smart" I can do instead? I suspect there may be a way to handle it more elegantly with residue techniques, but I am not fluent enough to know how, and I couldn't find a similar example in my go-to complex analysis texts.
Thanks!
EDIT: Also suppose none of the poles lie between $a$ and $b$; that should be reasonable based on my problem.
integration
integration
asked Jan 11 at 3:54
MathIsArtMathIsArt
1278
asked Jan 11 at 3:54
MathIsArtMathIsArt
1278
edited Jan 11 at 4:29
MathIsArt
asked Jan 11 at 3:54
MathIsArtMathIsArt
1278
asked Jan 11 at 3:54
MathIsArtMathIsArt
1278
asked Jan 11 at 3:54
MathIsArtMathIsArt
1278
1278
$begingroup$
I don't expect there would be a particular nice general form. If we denote the integrand by $f_i(x)$, then we have $f_{i + 1}(x) = frac{x}{x^2 + 2 gamma x + 1} f_i(x)$, which suggests trying to set up a recursion relation in $i$ and relating the partial fractions decompositions of $f_i, f_{i + 1}$.
$endgroup$
– Travis
Jan 13 at 23:18
$begingroup$
Thanks for your suggestion, I appreciate it. I will try this.
$endgroup$
– MathIsArt
Jan 14 at 1:20
add a comment |
$begingroup$
I don't expect there would be a particular nice general form. If we denote the integrand by $f_i(x)$, then we have $f_{i + 1}(x) = frac{x}{x^2 + 2 gamma x + 1} f_i(x)$, which suggests trying to set up a recursion relation in $i$ and relating the partial fractions decompositions of $f_i, f_{i + 1}$.
$endgroup$
– Travis
Jan 13 at 23:18
$begingroup$
Thanks for your suggestion, I appreciate it. I will try this.
$endgroup$
– MathIsArt
Jan 14 at 1:20
$begingroup$
I don't expect there would be a particular nice general form. If we denote the integrand by $f_i(x)$, then we have $f_{i + 1}(x) = frac{x}{x^2 + 2 gamma x + 1} f_i(x)$, which suggests trying to set up a recursion relation in $i$ and relating the partial fractions decompositions of $f_i, f_{i + 1}$.
$endgroup$
– Travis
Jan 13 at 23:18
$begingroup$
I don't expect there would be a particular nice general form. If we denote the integrand by $f_i(x)$, then we have $f_{i + 1}(x) = frac{x}{x^2 + 2 gamma x + 1} f_i(x)$, which suggests trying to set up a recursion relation in $i$ and relating the partial fractions decompositions of $f_i, f_{i + 1}$.
$endgroup$
– Travis
Jan 13 at 23:18
$begingroup$
Thanks for your suggestion, I appreciate it. I will try this.
$endgroup$
– MathIsArt
Jan 14 at 1:20
$begingroup$
Thanks for your suggestion, I appreciate it. I will try this.
$endgroup$
– MathIsArt
Jan 14 at 1:20
add a comment |
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$begingroup$
I don't expect there would be a particular nice general form. If we denote the integrand by $f_i(x)$, then we have $f_{i + 1}(x) = frac{x}{x^2 + 2 gamma x + 1} f_i(x)$, which suggests trying to set up a recursion relation in $i$ and relating the partial fractions decompositions of $f_i, f_{i + 1}$.
$endgroup$
– Travis
Jan 13 at 23:18
$begingroup$
Thanks for your suggestion, I appreciate it. I will try this.
$endgroup$
– MathIsArt
Jan 14 at 1:20