HPAS Maths Optional PYQs: Complex Analysis

On that page, you will find year-wise and question-wise solutions.

Based on your query, here is a compilation of all questions pertaining to Complex Analysis found within the provided HPAS Maths Optional Question Papers (2014–2024), presented in reverse chronological order.

HPAS 2024 Complex Analysis Questions

Question 1(b): Evaluate the contour integral \( \oint_{C} \frac{e^{3z}}{(z-\log 2)^4} dz \) Where C is the square with vertices at \(\pm 1, \pm i\). (This typically requires Cauchy’s Integral Formula for derivatives).

Question 4(b): Let G be a region and suppose that \(f:G \to \mathbb{C}\) is analytic such that \(f(G)\) is a subset of a circle. Then show that \(f\) is constant. (This relates to the open mapping theorem or general properties of analytic functions).

Question 8(a): Let C be the unit circle \(z = e^{i\theta}\). First, show that for any real constant \(a\), \(\int_{C} \frac{e^{az}}{z} dz = 2\pi i\). Then, write this integral in terms of \(\theta\) to derive the integration formula: \(\int_{0}^{\pi} e^{a \cos\theta} \cos(a \sin\theta) d\theta = \pi\). (This combines Cauchy’s Integral Formula with definite integral derivation).

HPAS 2023 Complex Analysis Questions

Question 5(a): Determine the analytic function \(f(z) = u+iv\) if \(u-v = \frac{\cos x + \sin x – e^{-y}}{2(\cos x – \cosh y)}\) and \(f\left(\frac{\pi}{2}\right) = 0\). (This uses the Milne-Thomson method or related techniques to construct an analytic function from a linear combination of its components).

Question 5(b): Show that the transformation \(w = z + \frac{1}{z}\) converts the straight line \(\text{arg}(z) = \alpha\) (where \(|\alpha| < \frac{\pi}{2}\)) into a branch of a hyperbola with eccentricity \(\sec\alpha\). (This is a question on conformal mapping).

HPAS 2021 Complex Analysis Questions

Question 1(b): Find a transformation \(w = f(z)\) which maps the real axis of the \(z\)-plane onto the real axis in the \(w\)-plane. (This involves properties of transformations, likely bilinear/Möbius).

Question 3(b): Using the concept of residue, determine the value of the integral: \(\int_{0}^{\infty} \frac{\sin(mx)}{x} dx\) when \(m > 0\). (This requires the application of the Cauchy Residue Theorem to evaluate a real improper integral).

HPAS 2020 Complex Analysis Questions

Question 3(c): Let \(f(z) = \frac{z-1}{z+1}\) and let L be the line in the z-plane through \(z=0\) and \(z=1+i\). Find the image of L under \(f\). (This involves complex mapping and transformations).

Question 4(a): Find the bilinear transformation which maps the points \(0, i, 1+i\) onto \(2i, -1, \text{ and } 0\), respectively.

Question 4(b): Prove that every bounded entire function is constant. Is \(f(z) = \sin z\) bounded on \(\mathbb{C}\)? Support your answer. (This directly tests Liouville’s Theorem and knowledge of standard complex functions).

HPAS 2019 Complex Analysis Questions

Question 1(b): Examine if the function \(u = e^{2xy}\sin(x^2 – y^2)\) is harmonic. Find the complex function \(f(z)\) in terms of z, where \(f(z) = u+iv\). (This requires checking the Laplace equation and constructing the analytic function).

Question 2(a): Find the transformation which maps the semi-infinite strip of width \(\pi\), bounded by the lines \(v=0\), \(v=\pi\), and \(u=0\), into the upper half of the \(z\)-plane. (This is a conformal mapping problem).

Question 2(b): Evaluate \(\int_C \frac{3z^2+z}{z^2-1} dz\) where C is the circle \(|z-1|=1\). (This requires the use of Cauchy’s Integral Formula or the Residue Theorem).

HPAS 2018 Complex Analysis Questions

Question 1(c): If \(X = \sqrt{-1}\), then find the value of \(X^X\). (This is a complex number arithmetic question).

HPAS 2017 Complex Analysis Questions

Question 5(a): Prove that the function \(u(x,y) = x^3 – 3xy^2\) is harmonic and obtain its conjugate. (This involves checking the Laplace equation and finding the harmonic conjugate \(v\)).

Question 5(b): Evaluate: \(\int_C \frac{e^{3z}}{z-\pi i} dz\) where C is the circle \(|z-1|=4\). (This requires checking if \(\pi i\) lies inside C and applying Cauchy’s Integral Formula).

HPAS 2016 Complex Analysis Questions

Question 1(b): Show that \(f(z)=|z|^{2}=x^{2}+y^{2}\) has a derivative at the origin. (This is a classic example testing the limits of the Cauchy-Riemann equations).

Question 2(b): Evaluate the integral: \(\int_{C}\sqrt{z}dz \text{, where C: } z=z(t)=e^{it}, 0\le t\le2\pi\). (This is a contour integration problem).

Question 4(b): By using Cauchy Residue theorem, evaluate the integral: \(\int_{0}^{\infty}\frac{x^{2}}{x^{6}+1}dx\). (Application of the Residue Theorem to real integrals).

Question 5(a): Determine the bilinear transformation that maps the points \(z=0, -i, 2i\) into the points \(w=5i, \infty, -i/3\) respectively. What is the image of \(|z|<1\) under this transformation?.

HPAS 2015 Complex Analysis Questions

Question 1(d): Form a partial differential equation by eliminating the functions from the equation: \(Z=f(x+iy)+\phi(x-iy)\), where \(i=\sqrt{(-1)}\). (This uses complex variables in the context of PDE formation).

Question 5(a): Verify Cauchy’s theorem for the function \(5 \sin 2z\) if C is the square with vertices \(1\pm i\) and \(-1\pm i\), where \(i=\sqrt{(-1)}\) and C: closed contour.

Question 5(b): Show that the transformation \(w=\frac{2z+3}{z-4}\) maps the circle \(x^{2}+y^{2}-4x=0\) into the straight line \(4u+3=0\).

HPAS 2014 Complex Analysis Questions

Question 5(a): If \(f(z)=u+iv\) is an analytic function of \(z=x+iy\) and \(u-v=e^{x}(\cos y-\sin y)\), find \(f(z)\) in terms of \(z\).

Question 5(b): If \(w=f(z)\) represents a conformal transformation of a domain D in the z-plane into a domain D’ of the w-plane, then prove that \(f(z)\) is an analytic function of \(z\) in D. (This links conformal mapping directly to the property of analyticity).