- #1
sahil_time
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Homework Statement
∮ dz/(2 - z*) over Curve |z|=1?
where z* = conjugate of z
How to solve this?
Homework Equations
I tried doing by taking z*=e^(-iθ) , the answer was zero
Then i did it by taking z*=1/z which gives ∏i/2.
sahil_time said:Homework Statement
∮ dz/(2 - z*) over Curve |z|=1?
where z* = conjugate of z
How to solve this?
Homework Equations
I tried doing by taking z*=e^(-iθ) , the answer was zero
Then i did it by taking z*=1/z which gives ∏i/2.
The Attempt at a Solution
HallsofIvy said:Why would you take z*= 1/z??
sahil_time said:Yes, i did take z*=e^(-iθ).
then ∫dz/(2-e^(-iθ)) = ∫ e^(iθ)/( 2e^(iθ) - 1)dz
taking 2e^(iθ) -1 = t
2ie^(iθ)dθ = dt
But limits 0 →2π change to 1→1
Therefore ans is zero!
What i think is that the function is not analytic, therefore we cannot
integrate over 0→2π directly. We must break the limits. For eg: if you
take 0→π and multiply the Whole Integral by 2, you will get the ans as
∏i/2. But i cannot see any reason as to why the function ceases to be
"non analytic".
Thankyou. :)
sahil_time said:Thanx alot! But can you suggest some good mathematical books which
explains these advanced concepts in proper detail.?
Thanx again! :)
The Cauchy Integral Theorem, also known as the Cauchy-Goursat Theorem, is a fundamental theorem in complex analysis that states that if a function is analytic inside and on a simple closed contour, then the integral of that function over the contour is equal to zero.
The Cauchy Integral Theorem was discovered by Augustin-Louis Cauchy, a French mathematician, in the early 19th century. However, it was later refined and expanded upon by other mathematicians such as Joseph-Louis Lagrange and Pierre-Simon Laplace.
The Cauchy Integral Theorem is significant because it allows for the evaluation of complex integrals without having to use complex techniques such as contour integration. It also has many applications in physics, engineering, and other fields that involve complex functions.
The Cauchy Integral Theorem is closely related to the Cauchy-Riemann equations, which are a set of necessary and sufficient conditions for a function to be analytic. In fact, the Cauchy Integral Theorem can be derived from the Cauchy-Riemann equations.
Yes, the Cauchy Integral Theorem can be extended to higher dimensions through the use of multivariate calculus and complex analysis. This allows for the evaluation of integrals over surfaces and volumes in more than two dimensions.