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Tangent87
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I have attached a pdf of my problem and attempted solution. I seem to be a factor of f'(z) out from the required solution, can anyone see where I've gone wrong?
The Complex Integral Residue Theorem is a mathematical concept in complex analysis that relates the value of a complex integral around a closed curve to the singularities of a function inside the curve. It allows for the calculation of difficult integrals using the residues, or the values of a function at its singular points.
The Complex Integral Residue Theorem is used in several areas of mathematics, including complex analysis, number theory, and differential equations. It is especially useful for calculating integrals that cannot be solved using traditional methods, as it provides a shortcut for finding their values.
In the context of the Complex Integral Residue Theorem, singularities refer to the points where a function is undefined or infinite. These points can be poles, essential singularities, or removable singularities, and they play a crucial role in the theorem's calculations.
The formula for the Complex Integral Residue Theorem is given by:
∫γ f(z) dz = 2πi ∑k Res(f,zk),
where ∫γ f(z) dz represents the complex integral around the closed curve γ, and Res(f,zk) represents the residue of the function f at the singular point zk.
The Complex Integral Residue Theorem has many applications in mathematics and physics, including calculating definite integrals, solving differential equations, and evaluating infinite series. It is also used in the study of complex functions and their properties, such as analyticity and meromorphicity.