Complex Analysis: Calculating the Limit of I(r)

[regularngon]

Some hints/help woudl be greatly appreciated!

Let I(r) = integral over gamma of (e^iz)/z where gamma: [0,pi] -> C is defined by gamma(t) = re^it. Show that lim r -> infinity of I(r) = 0.

 

[quasar987]

Well what work have you done so far?

The first step would be to write the integral with t as the variable of integration.

 

[Hurkyl]

It may matter how far into your complex analysis course you are... have you, for example, just learned the definition of such an integral, or have you learned other things too?

 

[regularngon]

I am only supposed to assume the definition of the integral, which is why I'm stuck.

 

[quasar987]

I'm not sure I see where the trouble is. I would just write the integral in simplified form, then bring the limit into the integral after verifying that the convergence on [0,pi] is uniform.

 

[mathwonk]

do you know how e^w behaves geometrically? thnink about what e^(iz) does to points z on the upper half of a circle of radius r.

first where does iz live if z is on such a semicircle?

second, where does e^w send those points iz?

then what happens when you divide by z?

you only need to understand the size of the integrand here.

so nothing big seems required here, no uniform convergence or anything.

just a basic estimate ofn the size of an integral in terms of the size of the integrand and the path.

you have to check me of course on this, as i am doing this in my head immediately after waking up, no coffee yet or anything.