Contour Integration: Branch cuts

[WWCY]

Homework Statement

I am supposed to evaluate the contour integral of the positive branch of $z^{-1/2}$ over the following contour:

I believe the answer should be 0, by Cauchy's theorem (loop encloses no poles), but my methods of parameterization have led to non-zero answers.

Homework Equations

The Attempt at a Solution

Here are the following parameterizations I have tried so far,

A) Loop parameterized by $z = R_A e^{i\theta}$ from $\theta = -\pi$ to $\pi$
B) Line parameterized by $z = Re^{i\pi}$ from $-R_A$ to $-R_C$
C) Loop parameterized by $z = R_C e^{i\theta}$ from $\theta = \pi$ to $-\pi$
D) Line parameterized by $z = Re^{-i\pi}$ from $-R_C$ to $-R_A$

I believe that I am actually doing integrals on the cut by using this method (hence an incorrect answer), but I can't seem to find a way to avoid the cut and am therefore stuck.

Any advice is greatly appreciated

 

[andrewkirk]

To stay away from the cut, the first arc mustn't go from $-\pi$ to $\pi$ but from $\epsilon-\pi$ to $\pi-\epsilon$, where $\epsilon$ is a small angle.
Then you can parameterise B and D as horizontal lines at heights $\pm y_B$ that are trig functions of $R_A$ and $\epsilon$ and starting and ending at horizontal coordinates $x_A,x_C$ that are respectively functions of $(R_A,\epsilon)$ and $(R_C,y_B)$. Finally, you need to work out a new angle $\psi$ that is a function of $(R_C,y_B)$ so that $\pm \psi$ are the integration limits for curve C.

Write a formula for the total integral, which will be a function of $R_A,R_C$ and $\epsilon$. Then work out the limit of that formula as $\epsilon\to 0$.

EDIT: We can simplify the process by getting rid of the parameter $R_C$ and instead having the loop C go around the origin at radius $y_B$, joining to the lines B and D tangentially rather than with a kink. Then $x_C=0$ so that loop C and lines B, D all end neatly on the y axis.

 

[WWCY]

To stay away from the cut, the first arc mustn't go from $-\pi$ to $\pi$ but from $\epsilon-\pi$ to $\pi-\epsilon$, where $\epsilon$ is a small angle.
Then you can parameterise B and D as horizontal lines at heights $\pm y_B$ that are trig functions of $R_A$ and $\epsilon$ and starting and ending at horizontal coordinates $x_A,x_C$ that are respectively functions of $(R_A,\epsilon)$ and $(R_C,y_B)$. Finally, you need to work out a new angle $\psi$ that is a function of $(R_C,y_B)$ so that $\pm \psi$ are the integration limits for curve C.

Write a formula for the total integral, which will be a function of $R_A,R_C$ and $\epsilon$. Then work out the limit of that formula as $\epsilon\to 0$.

EDIT: We can simplify the process by getting rid of the parameter $R_C$ and instead having the loop C go around the origin at radius $y_B$, joining to the lines B and D tangentially rather than with a kink. Then $x_C=0$ so that loop C and lines B, D all end neatly on the y axis.

Thanks very much! I'll give it a shot. Appreciate the help.

Would I be right in saying that working in terms of $\pi$ instead of $\pi + \epsilon$ means working on the cut?

 

[andrewkirk]

Would I be right in saying that working in terms of $\pi$ instead of $\pi + \epsilon$ means working on the cut?

Yes.