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hancock.yang@
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Homework Statement
Using the Cauchy Integral Formula compute the following integrals,where C is a circle of radius 2a centered at z=o, where 2a<pi
Homework Equations
[tex]\oint\frac{(z-a)e^{z}}{(z+a)sinz}[/tex]
yes.Dick said:Integrated around |z|=2a? If you write your integrand as f(z)/z (you figure out what f(z) would need to be) the Cauchy integral formula would tell you what the integral is in terms of f(0). Which would be very nice but the Cauchy integral formula doesn't apply because f(z)/z isn't holomorphic inside |z|=2a. Aside from a removable singularity at z=0 there's a pole at z=a. Are you sure you wrote the problem down correctly?
hancock.yang@ said:yes.
I have sperate the form like this:
[tex]\oint\frac{(z-a)e^{z}}{(z+a)}\frac{dz}{sinz}[/tex]+[tex]\oint\frac{(z-a)e^{z}}{sinz}\frac{dz}{(z-a)}[/tex]
The original problem is this:Dick said:I really don't understand what you are saying there. Did you alter the original problem statement? If so, what was the original?
Dick said:Ok, so you are really doing as a residue theorem problem, not just a Cauchy integral problem. In one of those integrals you should multiply by (z-0) and let z approach 0 and in the other one multiply by (z+a) and let z approach -a, right?
Dick said:You should probably look it up. I don't necessarily explain things that well. The Cauchy Integral Theorem just says f(a)=(1/(2*pi*i)) times the contour integral f(z)/(z-a) over a circle where f(z) is holomorphic. The residue theorem is the obvious generalization of that to the case where you have multiple poles in a single domain and you cut out a circle around each one and add them up. Which is what you are doing.
The Cauchy Integral Formula is a fundamental theorem in complex analysis that relates the values of a function on the boundary of a region to its values inside the region. It is used to calculate the values of a holomorphic function at any point, given its values on the boundary of a disk or contour.
The Cauchy Integral Formula was discovered by French mathematician Augustin-Louis Cauchy in the early 19th century. He was one of the pioneers of complex analysis and made significant contributions to the field.
The Cauchy Integral Formula is significant because it allows for the evaluation of complex integrals without having to directly calculate them. It also provides a way to extend the concept of differentiation to complex functions, which has important applications in physics and engineering.
No, the Cauchy Integral Formula only applies to holomorphic functions, which are complex functions that are differentiable at every point in their domain. If a function is not holomorphic, the Cauchy Integral Formula cannot be used to evaluate its values.
The Cauchy Integral Formula has a wide range of applications, including solving boundary value problems in physics and engineering, calculating electric potentials and currents in electromagnetism, and finding solutions to differential equations. It is also used in the study of fluid dynamics and mathematical physics.