- #1
demoncore
- 18
- 1
Missing homework template due to originally being posted in other forum.
I am attempting to calculate the following integral.
$$\frac{1}{2\pi i}\int_C \frac{du}{u^2} exp({-\frac{(q - \frac{q_0}{2i} (u - u^{-1}))^2}{2\sigma^2}})$$
Taken over the unit disk. I first make the substitution $$z = q - \frac{q_0}{2i} (u - u^{-1})$$ So,
$$dz = -\frac{q_0}{2i}(1 + u^{-2})du$$
When I attempt to back-substitute in for u, however, I find the following expression:
$$u = \frac{(q - z)i \pm \sqrt{q_0^2 - (q - z)^2}}{q_0}$$
I am not sure where to proceed from here. Arbitrarily choosing one or the other solution for u doesn't seem to give me the correct answer. Any help would be appreciated.
$$\frac{1}{2\pi i}\int_C \frac{du}{u^2} exp({-\frac{(q - \frac{q_0}{2i} (u - u^{-1}))^2}{2\sigma^2}})$$
Taken over the unit disk. I first make the substitution $$z = q - \frac{q_0}{2i} (u - u^{-1})$$ So,
$$dz = -\frac{q_0}{2i}(1 + u^{-2})du$$
When I attempt to back-substitute in for u, however, I find the following expression:
$$u = \frac{(q - z)i \pm \sqrt{q_0^2 - (q - z)^2}}{q_0}$$
I am not sure where to proceed from here. Arbitrarily choosing one or the other solution for u doesn't seem to give me the correct answer. Any help would be appreciated.