- #1
dipole
- 555
- 151
I'd like to evaluate the integral,
[tex] \int^{i\infty}_{-i\infty} \frac{e^{iz}}{z^2 + a^2}dz [/tex]
along the imaginary axis. This function has poles at [itex] z = \pm ia [/itex], with corresponding residues [itex] \textrm{res}(\frac{e^{iz}}{z^2 + a^2},\pm ia) = \pm\frac{e^{\mp a}}{2ai} [/itex]
My question is - I'm not sure what contour to use. If I go from a segment [itex] (-iR, iR) [/itex], while skirting around the poles, and close it with a semi-circle in the right-half plane the resulting arc in the lower-right half plane will not vanish according to Jordan's lemma... I can't find any examples in my book about how to do contours along the imaginary axis - they all go along the real axis and mostly make use of Jordan's Lemma to simplify things, which doesn't seem applicable here.
Any suggestions?
[tex] \int^{i\infty}_{-i\infty} \frac{e^{iz}}{z^2 + a^2}dz [/tex]
along the imaginary axis. This function has poles at [itex] z = \pm ia [/itex], with corresponding residues [itex] \textrm{res}(\frac{e^{iz}}{z^2 + a^2},\pm ia) = \pm\frac{e^{\mp a}}{2ai} [/itex]
My question is - I'm not sure what contour to use. If I go from a segment [itex] (-iR, iR) [/itex], while skirting around the poles, and close it with a semi-circle in the right-half plane the resulting arc in the lower-right half plane will not vanish according to Jordan's lemma... I can't find any examples in my book about how to do contours along the imaginary axis - they all go along the real axis and mostly make use of Jordan's Lemma to simplify things, which doesn't seem applicable here.
Any suggestions?