- #1
juan.
- 8
- 0
Hello everyone! I'm having some troubles when I try to solve improper integrals exercises that have singularities on the real axis. I have made a lot of exercises where singularities are inside a semicircle in the upper half side, but I don't know how to solve them when the singularities are on the real axis.
I read some books but I think they are not very good explained (at least, I can't understood them).
This is the exercise:
[tex]\int_{-\infty}^{\infty} \frac{cos(2\pi x)}{x^2-1} dx[/tex]
Using complex variable, I have:
[tex]f(z) =\frac{exp(i2\pi z)}{z^2-1}[/tex]
so there are 2 singularities:
[tex]z_1 = -1[/tex] and [tex]z_2 = 1[/tex]
I use a curve [tex]C[/tex] that is holomorphic inside it, because both singularities are out of it. Of course, I can divide [tex]C[/tex] in 6 curves: [tex]C_R[/tex] that is the "roof" of the curve and, using Jordan's Lemma, I can prove that
[tex]\int_{C_R}^{ } \frac{exp(i2\pi z)}{z^2-1} dx = 0[/tex]
but I don't know what do I have to do now. I saw in some places they said that the Residue Theorem over the semicircle around the singularities was something like [tex]-i\pi\sum{}{}Res[f(z), z_k][/tex] but I didn't understand why.
I hope you can help me, because I don't know what can I do.
Thanks!
I read some books but I think they are not very good explained (at least, I can't understood them).
This is the exercise:
[tex]\int_{-\infty}^{\infty} \frac{cos(2\pi x)}{x^2-1} dx[/tex]
Using complex variable, I have:
[tex]f(z) =\frac{exp(i2\pi z)}{z^2-1}[/tex]
so there are 2 singularities:
[tex]z_1 = -1[/tex] and [tex]z_2 = 1[/tex]
I use a curve [tex]C[/tex] that is holomorphic inside it, because both singularities are out of it. Of course, I can divide [tex]C[/tex] in 6 curves: [tex]C_R[/tex] that is the "roof" of the curve and, using Jordan's Lemma, I can prove that
[tex]\int_{C_R}^{ } \frac{exp(i2\pi z)}{z^2-1} dx = 0[/tex]
but I don't know what do I have to do now. I saw in some places they said that the Residue Theorem over the semicircle around the singularities was something like [tex]-i\pi\sum{}{}Res[f(z), z_k][/tex] but I didn't understand why.
I hope you can help me, because I don't know what can I do.
Thanks!