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Hi!

I am relatively new to complex analysis and I am trying to write down the following integral in Sage Math:

$$ I(k) = \frac{1}{2i\pi}\oint\frac{(1-t^2)}{(1-t)^n}\frac{dz}{t^{k+1}} $$

from a paper that can be found at: http://magali.bardet.free.fr/Publis/ltx43BF.pdf

whereby $$S(n) = \frac{(1-t^2)}{(1-t)^n} $$ is a formal power series. The Cauchy Integral will produce the k-th coefficient of $S(n)$. I tried doing the following:

def deg_reg_Cauchy(k, n, m):
    R.<t> = PowerSeriesRing(CC, 't')
    constant_term = 1/(2*I*pi)
    s = (1-t**2)**m / (t**(k+1)*(1-t)**n)
    s1 = constant_term * s.integral()
    return s1

I realize this is probably very wrong and I used $0$ till $2\pi$ as simple placeholders until I find appropriate values. Does anyone have any tips on how to go about this, please? Below is the error message that is being outputted by Sage.

ArithmeticError: The integral of is not a Laurent series, since t^-1 has nonzero coefficient.

Thank you!

Hi!

I am relatively new to complex analysis and I am trying to write down the following integral in Sage Math:

$$ I(k) = \frac{1}{2i\pi}\oint\frac{(1-t^2)}{(1-t)^n}\frac{dz}{t^{k+1}} \frac{1}{2i\pi}\oint\frac{(1-t^2)}{(1-t)^n}\frac{dt}{t^{k+1}} $$

from a paper that can be found at: http://magali.bardet.free.fr/Publis/ltx43BF.pdf

whereby $$S(n) = \frac{(1-t^2)}{(1-t)^n} $$ is a formal power series. The Cauchy Integral will produce the k-th coefficient of $S(n)$. I tried doing the following:

def deg_reg_Cauchy(k, n, m):
    R.<t> = PowerSeriesRing(CC, 't')
    constant_term = 1/(2*I*pi)
    s = (1-t**2)**m / (t**(k+1)*(1-t)**n)
    s1 = constant_term * s.integral()
    return s1

I realize this is probably very wrong and I used $0$ till $2\pi$ as simple placeholders until I find appropriate values. Does anyone have any tips on how to go about this, please? Below is the error message that is being outputted by Sage.

ArithmeticError: The integral of is not a Laurent series, since t^-1 has nonzero coefficient.

Thank you!

Hi!

I am relatively new to complex analysis and I am trying to write down the following integral in Sage Math:

$$ I(k) = \frac{1}{2i\pi}\oint\frac{(1-t^2)}{(1-t)^n}\frac{dt}{t^{k+1}} $$

from a paper that can be found at: http://magali.bardet.free.fr/Publis/ltx43BF.pdf

The contour is a unit circle around the origin with a radius less than 1.

whereby $$S(n) = \frac{(1-t^2)}{(1-t)^n} $$ is a formal power series. The Cauchy Integral will produce the k-th coefficient of $S(n)$. I tried doing the following:

def deg_reg_Cauchy(k, n, m):
    R.<t> = PowerSeriesRing(CC, 't')
    constant_term = 1/(2*I*pi)
    s = (1-t**2)**m / (t**(k+1)*(1-t)**n)
    s1 = constant_term * s.integral()
    return s1

I realize this is probably very wrong and I used $0$ till $2\pi$ as simple placeholders until I find appropriate values. Does anyone have any tips on how to go about this, please? Below is the error message that is being outputted by Sage.

ArithmeticError: The integral of is not a Laurent series, since t^-1 has nonzero coefficient.

Thank you!

Hi!

I am relatively new to complex analysis and I am trying to write down the following integral in Sage Math:

$$ I(k) = \frac{1}{2i\pi}\oint\frac{(1-t^2)}{(1-t)^n}\frac{dt}{t^{k+1}} $$

from a paper that can be found at: http://magali.bardet.free.fr/Publis/ltx43BF.pdf

The contour is a unit circle around the origin with a radius less than 1.

whereby $$S(n) = \frac{(1-t^2)}{(1-t)^n} $$ is a formal power series. The Cauchy Integral will produce the k-th coefficient of $S(n)$. I tried doing the following:

def deg_reg_Cauchy(k, n, m):
    R.<t> = PowerSeriesRing(CC, 't')
    constant_term = 1/(2*I*pi)
    s = (1-t**2)**m / (t**(k+1)*(1-t)**n)
    s1 = constant_term * s.integral()
    return s1

I realize this is probably very wrong and I used $0$ till $2\pi$ as simple placeholders until I find appropriate values. Does anyone have any tips on how to go about this, please? Below is the error message that is being outputted by Sage.

ArithmeticError: The integral of is not a Laurent series, since t^-1 has nonzero coefficient.

Thank you!