Complex Analysis Complex Integration Question

[ha9981]

Its question 1(g) in the picture. My work is shown there as well. This has to do with independence of path of a contour. Reason I am suspicious is that first there is a different answer online and second it says "principal branch" which I have not understood. Does that mean a straight line for start to end point?

The contour starts at pi and ends at i on the complex plane. Also the function integrated is z^1/2.

Here is the picture:

sorry idk why image is flipped side ways when uploading, i fear this will make it even more challenging for someone to want to answer this question.

 

[Skrew]

Principle branch normally refers to Log, are you sure you are looking at the right problem?

Also what textbook are you using? I'm curious.

Ignore the above, I forgot how fractional exponents are defined.

 

[Dickfore]

Principle branch normally refers to Log

This isn't true. There is a principal branch for the square root as well.

 

[Skrew]

This isn't true. There is a principal branch for the square root as well.

You're right, I forgot how fractional exponets were defined(as it involves log..).

 

[ha9981]

Textbook is Fundamentals of Complex Analysis with Applications to Engineering and Science by E. B. Saff & A. D. Snider. Also what has the principal have to do with fractional exponents? Is there a region where it isn't defined?

 

[Skrew]

The antiderivative of z^1/2 = 2/3*z^3/2.

Now 2/3*z^3/2 = 2/3*e^ln(z^3/2) = 2/3*e^(3/2*ln(z)) (if I remmember correctly)

Now the fundamental theoreom of calculus in the complex case says that the path is independent in any domain where an antiderivativie exists.

The principle branch of ln is the complex plane excluding the orgin and the negative axis.

 

[ha9981]

So the contour in the diagram should have no interference with the principal branch right?