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## Main Question or Discussion Point

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!!