definite complex integrals ..help me plz

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adrian zizo
adrian zizo 2015-3-14
评论： Torsten 2022-1-18
how ca I answer this integration by using matlab
∫(x^3+3)/((x^2+1)(x^2+4)) interval[-∞,∞] ?
I answer it by using calculation by using residue :
∴(z^2+1)(z^2+4)=(z-j)(z+j)+(z-j2)(z+j2)
z = +j
R_1=(-1+3)/((+j+j)(-1+4))=-1/3 j
z =j2
R_2=((-4)+3)/((-4+1)(j2+j2))=-1/12 j
I=-5/12 J(2πJ)=5/6 π

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回答（1 个）

Roger Stafford
Roger Stafford 2015-3-14
Your integral from minus infinity to plus infinity is divergent, because for large x the integrand behaves like 1/x, which would give log(x) as an integral, and log(x) becomes infinite. In other words, there is a non-zero residue at infinity and you are integrating through infinity, so the integral is not well-defined. It is as though you were integrating through a simple pole singularity on the finite complex plane, which would be ill-defined.
  2 个评论
Matthew Young
Matthew Young 2022-1-18
Obviously it's too late now but the integral is convegent.
you can split it into \int x^3 / ((x^2 +1)(x^2+4)) + \int 3 / ((x^2 +1)(x^2+4)). The first term evaluates to 0 and the second term behaves lke 1/x^4 . You can split the second term up using partial fractions and then use arctan to evaluate the integral.
Torsten
Torsten 2022-1-18
This is only true if you mean the Cauchy Principal Value of the integral.
In the usual sense, the integral does not exist as @Roger Stafford noticed correcty.

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