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Frequently Asked Questions
Q. 1) Define an analytic function. How does it differ from a differentiable function in the context of complex analysis?
An analytic function is differentiable in a region of the complex plane, and its derivative is also continuous. A differentiable function may not be analytic unless it satisfies the Cauchy-Riemann equations.
Q. 2) What is a singularity of a complex function? Classify the singularities as removable singularities, poles, and essential singularities with examples.
A singularity is a point where a function is not analytic. A removable singularity allows the function to be extended, e.g., ???? ( ???? ) = sin ( ???? ) ???? f(z)= z sin(z) at ???? = 0 z=0. A pole is a point where the function goes to infinity, e.g., ???? ( ???? ) = 1 ???? f(z)= z 1 at ???? = 0 z=0. An essential singularity has a non-removable, wild behavior, e.g., ???? ( ???? ) = ???? 1 / ???? f(z)=e 1/z at ???? = 0 z=0.
Q. 3) What is the concept of a residue in complex analysis? How is the residue of a function at a singularity important for evaluating contour integrals?
The residue of a function at a singularity is the coefficient of 1 ???? z 1 in its Laurent series. The residue is used in the residue theorem to evaluate contour integrals.
Q. 4) Explain the concept of a Laurent series. How does it differ from a Taylor series, and in which situations is it used in complex analysis?
A Laurent series is a representation of a complex function as a series with both positive and negative powers of ???? z. It is used to express functions around singularities, unlike the Taylor series, which only involves non-negative powers.
Q. 5) State and prove Cauchy's Integral Theorem. How is it used to evaluate integrals in complex analysis?
Cauchy's Integral Theorem states that the integral of a holomorphic function over a closed contour is zero. It simplifies complex integrals by reducing them to integrals over simpler contours or regions.
Q. 6) Discuss the concept of a conformal map in complex analysis. What properties must a function satisfy to be considered conformal?
A conformal map preserves angles and shapes locally. The function must be holomorphic and have a non-zero derivative at every point in its domain.
Q. 7) What is the residue theorem? How does it relate to contour integration, and how can it be used to compute real integrals?
The residue theorem states that the integral of a function around a closed contour is 2 ???? ???? 2πi times the sum of residues inside the contour. It is used to compute real integrals by transforming them into contour integrals.
Q. 8) Explain the concept of a branch cut in complex analysis. How does it help in defining multi-valued functions like
log
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log(z) or
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1
/
2
z
1/2
?
A branch cut is a curve in the complex plane where a multi-valued function, such as the logarithm, is discontinuous. It is used to define a single-valued branch of the function by restricting its domain
Q. 9) State and explain the maximum modulus principle. How can this principle be used to determine the behavior of a function inside a domain?
The maximum modulus principle states that if a function is analytic and non-constant in a domain, its modulus attains its maximum on the boundary of the domain. It is used to bound the behavior of functions inside the domain.
Q. 10) What is a biholomorphic function? Give an example and explain why it is important in complex analysis.
A biholomorphic function is a bijective, holomorphic function with a holomorphic inverse. It is important because it preserves the structure of complex manifolds, acting like an isomorphism in complex analysis.
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