Hello! Sorry, if I choosed wrong thread, but I really couldn't find a thread about complex analysis.
So, the task is the following:
. . . . .\(\displaystyle \large{\displaystyle 4.\, v.p.\, \int_{-\infty}^{+\infty}\, \dfrac{1\, -\, e^{i \alpha x}}{x^2}\, dx\, (\alpha\, <\, 0)}\)
Let's be f(z) = (1-exp(iaz))/z^2 . This function has only one singularity in z = 0. And it is pole with order of 2.
As I understand, I should use contour like half of a ring or half of a circle.
In the way using half of a circle with radius R, there is a singularity z = 0 placed right on the countur . I would like to use Cauchy's theorem and lemma about singularities on a contour, but the order of pole is 2.
Maybe I can avoid this thing about 2d order of pole? Or, maybe I am wrong about the v.p.
So, could You help me please? Thank you!
So, the task is the following:
. . . . .\(\displaystyle \large{\displaystyle 4.\, v.p.\, \int_{-\infty}^{+\infty}\, \dfrac{1\, -\, e^{i \alpha x}}{x^2}\, dx\, (\alpha\, <\, 0)}\)
Let's be f(z) = (1-exp(iaz))/z^2 . This function has only one singularity in z = 0. And it is pole with order of 2.
As I understand, I should use contour like half of a ring or half of a circle.
In the way using half of a circle with radius R, there is a singularity z = 0 placed right on the countur . I would like to use Cauchy's theorem and lemma about singularities on a contour, but the order of pole is 2.
Maybe I can avoid this thing about 2d order of pole? Or, maybe I am wrong about the v.p.
So, could You help me please? Thank you!
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