- #1

- 19

- 0

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.

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter regularngon
- Start date

- #1

- 19

- 0

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.

- #2

quasar987

Science Advisor

Homework Helper

Gold Member

- 4,783

- 18

Well what work have you done so far?

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

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

Last edited:

- #3

Hurkyl

Staff Emeritus

Science Advisor

Gold Member

- 14,916

- 19

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?

Last edited:

- #4

- 19

- 0

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

- #5

quasar987

Science Advisor

Homework Helper

Gold Member

- 4,783

- 18

- #6

mathwonk

Science Advisor

Homework Helper

2020 Award

- 11,120

- 1,319

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.

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.

Last edited:

Share: