Another Complex Analysis Question

[Edwin]

Suppose you have a Meromorphic function f(z) that has a zero at some point in the complex plane. Suppose you create two parallel contours Y1 and Y2 that are parallel and infinitely close to each other yet still contains the zero (the contours are infinitely close to the zero but don't run through the zero). What effects, or symmetry properties does the proximity of the contours to each other yield on the "change in argument of the function" as you traverse the two contours that contain the zero?

Inquisitively,

Edwin G. Schasteen

 

[shmoe]

What does "infinitely close to the zero" mean? In any case integrating on or near a zero isn't a problem. Maybe you meant a pole?

There are results about the intergral over an arc of a circle centered at a residue as the size of the circle tends to zero. Under the right conditions you'll end up getting the usual contribution of the residue times the angle/2pi. eg. if you were to integrate along the x-axis and avoid a pole at 0 by a half circle centered at 0 lying above the x-axis, as this half circle's radius tends to 0 you get half the residue. I can't recall the exact conditions required off hand, but this should be in any complex analysis text.

 

[Edwin]

Oops! Yep, I meant pole. Thanks. I just wanted to make sure it would work for straight line contours close infinitely close to poles.

Best Regards,

Edwin