- #1
ArcanaNoir
- 779
- 4
Homework Statement
My class is working through chapter 2 of Newman's Analytic Number Theory text (on partitions). We have come to a part where he states that "elementary geometry gives the formula" (for the length of arc A) [tex] 4r\text{arcsin}\frac{\sqrt(2)(1-r)}{\sqrt(r)} [/tex]
We are attempting to find an integral over the curve |z|=r, where r<1, and specifically at this moment we want the length of the arc of our curve where |x|=r and [tex] \frac{|1-z|}{1-|z|}\le 3.[/tex]
I have attached a picture from someone's master's thesis that shows the curves in question. The circle inside the unit circle is the circle |z|=r, and the circle on the right I believe to be the circle |z-1|=3(1-r). We need the length of the arc of |z|=r that is inside the right hand circle.
Homework Equations
Law of sines, law of cosines, any trig identities, Pythagorean theorem, any trig/geometry.
The Attempt at a Solution
I have tried making triangles every which way to no avail and my professor was unable to resolve this problem in the half-hour or so that he worked on it with me. We will be going through this part of the chapter in class soon and I will be presenting. I'm not expected to be able to explain this but it would be nice for everyone in the class if we could see how it is done.
I came sort of close by pretending that the circle |z|=r intersected the center of the right hand circle. Using the law of sines and trig identities I got that the length of the arc in question would be [tex] 4r\text{arcsin}\frac{3(1-r)}{2r} [/tex]
It is also possible that the given solution contains some error, as this textbook is notoriously error-ridden.