Complex Analysis – Mathematics Optional PYQs (2008–2026)

Here are all 67 Complex Analysis questions asked in UPSC Mathematics Optional Paper II from 2008 to 2026, arranged under the 4 topics of the official syllabus. Within each topic the latest questions come first.

Each question ends with a code such as [2024/10m/1a] – the year, the marks and the question number in the paper.

3.1] Analytic functions

17 questions

1. Show that the function $f\left(z\right)=\sqrt{\left|xy\right|}$ is not analytic at the origin, although Cauchy-Riemann equations are satisfied at that point. [2026/10m/1d]

2. If $w=f\left(z\right)$ is an analytic function of $z$, then show that $(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2})log\left|f\prime \left(z\right)\right|=0$. [2024/10m/1b]

3. If $\varphi$ and $\psi$ are functions of $x$ and $y$ satisfying Laplace equation, then show that $f\left(z\right)=p+iq$, $i=\sqrt{-1}$ is an analytic function, where $p=\frac{\partial \varphi }{\partial y}-\frac{\partial \psi }{\partial x}$ and $q=\frac{\partial \varphi }{\partial x}+\frac{\partial \psi }{\partial y}$. [2024/10m/1d]

4. State the sufficient conditions for a function $f\left(z\right)=f(x+iy)=u(x,y)+iv(x,y)$ to be analytic in its domain. Hence, show that $f\left(z\right)=logz$ is analytic in its domain and find $\frac{df}{dz}$. [2023/10m/1d]

5. Prove that $u(x,y)=e^x(xcosy-ysiny)$ is harmonic. Find its conjugate harmonic function $v(x,y)$ and express the corresponding analytic function $f\left(z\right)$ in terms of $z$. [2023/15m/3b]

6. If $f\left(z\right)=u+iv$ is an analytic function of $z$, and $u-v=\frac{cosx+sinx-e^{-y}}{2cosx-e^y-e^{-y}}$, then find $f\left(z\right)$ subject to the condition $f\left(\frac{\pi }{2}\right)=0$. [2022/10m/1b]

7. If $v(r,\theta )=(r-\frac{1}{r})sin\theta$, $r\ne 0$, then find an analytic function $f\left(z\right)=u(r,\theta )+iv(r,\theta )$ [2020/15m/4a]

8. Suppose $f\left(z\right)$ is analytic function on a domain $D$ in $ℂ$ and satisfies the equation $Imf\left(z\right)={\left(Ref\left(z\right)\right)}^2$, $Z\in D$. Show that $f\left(z\right)$ is constant in $D$. [2019/10m/1d]

9. Prove that the function : $u(x,y)={(x-1)}^3-3x{y}^2+3y^2$ is harmonic and find its harmonic conjugate and the corresponding analytic function $f\left(z\right)$ in terms of $z$. [2018/10m/1c]

10. Let $f=u+iv$ be an analytic function on the unit disc $D=\{z\in ℂ:\left|z\right|<1\}$. Show that $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0=\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}$ at all points of $D$. [2017/15m/3b]

11. Is $v(x,y)=x^3-3x{y}^2+2y$ a harmonic function ? Prove your claim. If yes, find its conjugate harmonic function $u(x,y)$ and hence obtain the analytic function whose real and imaginary parts are $u$ and $v$ respectively. [2016/10m/1d]

12. Show that the function $v(x,y)=ln(x^2+y^2)+x+y$ is harmonic. Find its conjugate harmonic function $u(x,y)$. Also, find the corresponding analytic function $f\left(z\right)=u+iv$ in terms of $z$. [2015/10m/1d]

13. Prove that the function $f\left(z\right)=u+iv$, where $f\left(z\right)=\frac{x^3(1+i)-y^3(1-i)}{x^2+y^2},z\ne 0;f\left(0\right)=0$ satisfies Cauchy-Riemann equations at the origin, but the derivative of $f$ at $z=0$ does not exist. [2014/10m/1c]

14. Show that the function defined by $f\left(z\right)=\{\begin{array}{cc}\hfill \frac{x^3{y}^5(x+iy)}{x^6+y^{10}},\hfill & \hfill z\ne 0\hfill \\ \hfill 0,\hfill & \hfill z=0\hfill \end{array}$ is not analytic at the origin though it satisfies Cauchy-Riemann equations at the origin. [2012/12m/1c]

15. If $f\left(z\right)=u+iv$ is an analytic function of $z=x+iy$ and $u-v=\frac{e^y-cosx+sinx}{coshy-cosx}$, find $f\left(z\right)$ subject to the condition, $f\left(\frac{\pi }{2}\right)=\frac{3-i}{2}$. [2011/12m/1c]

16. Show that $u(x,y)=2x-x^3+3x{y}^2$ is a harmonic function. Find a harmonic conjugate of $u(x,y)$. Hence find the analytic function $f$ for which $u(x,y)$ is the real part. [2010/12m/1e]

17. (i) Evaluate the line integral ${\int }_c^{}f\left(z\right)dz$ where $f\left(z\right)=z^2$, $c$ is the boundary of the triangle with vertices $A(0,0)$, $B(1,0)$, $C(1,2)$ in that order. (ii) Find the image of the finite vertical strip $R:x=5$ to $x=9$, $-\pi \le y\le \pi$ of $z$-plane under exponential function. [2010/15m/4a]

3.2] Cauchy’s theorem and integral formula

8 questions

1. Find the function which is analytic inside and on the circle $C:z=e^{i\theta }$, $0\le \theta \le 2\pi$ and has the value $\frac{(a^2-1)cos\theta +i(a^2+1)sin\theta }{a^4-2a^{2}cos2\theta +1},$ on the circumference of $C$, where $a^2>1$. [2024/20m/2c]

2. Evaluate the integral ${\int }_C^{}(z^2+3z)dz$ counterclockwise from $(2,0)$ to $(0,2)$ along the curve $C$, where $C$ is the circle $\left|z\right|=2$. [2020/10m/1d]

3. Evaluate the integral ${\int }_c^{}Re\left(z^2\right)dz$ from 0 to $2+4i$ along the curve $C$ where $C$ is a parabola $y=x^2$. [2019/10m/3c]

4. Use Cauchy integral formula to evaluate ${\int }_c^{}\frac{e^{3z}}{{(z+1)}^4}dz,$ where $c$ is the circle $\left|z\right|=2$. [2012/15m/2c]

5. If the function $f\left(z\right)$ is analytic and one valued in $|z-a|<R$, prove that for $0<r<R$, $f\prime \left(a\right)=\frac{1}{\pi r}{\int }_0^{2\pi }P\left(\theta \right)e^{-i\theta }d\theta ,$ where $P\left(\theta \right)$ is the real part of $f(a+r{e}^{i\theta })$. [2011/15m/2c]

6. (i) Evaluate the line integral ${\int }_c^{}f\left(z\right)dz$ where $f\left(z\right)=z^2$, $c$ is the boundary of the triangle with vertices $A(0,0)$, $B(1,0)$, $C(1,2)$ in that order. (ii) Find the image of the finite vertical strip $R:x=5$ to $x=9$, $-\pi \le y\le \pi$ of $z$-plane under exponential function. [2010/15m/4a]

7. Evaluate ${\int }_C^{}[\frac{e^z}{z^2(z^2+2z+2)}+log(z-6)+\frac{1}{{(z-4)}^2}]dz$ where $C$ is the circle $\left|z\right|=3$. State the theorems you use in evaluating above integral. [2008/15m/4a]

8. Let $f\left(z\right)$ be an entire function satisfying $\left|f\left(z\right)\right|\le k{\left|z\right|}^2$ for some positive constant $k$ and all $z$. Show that $f\left(z\right)=a{z}^2$ for some constant $a$. [2008/15m/4b]

3.3] Power, Taylor and Laurent series

18 questions

1. Prove that $cosh(z+\frac{1}{z})=a_0+{\sum }_{n=1}^{\infty }{a}_n(z^n+\frac{1}{z^n}),$ where $a_n=\frac{1}{2\pi }{\int }_0^{2\pi }cosn\theta cosh(2cos\theta )d\theta ,\mathrm{\text{for}}n=0,1,2,\dots .$ [2026/15m/3a]

2. Expand $f\left(z\right)=\frac{1}{(z+1)(z+3)}$ in a Laurent series valid for $1<\left|z\right|<3$. [2025/10m/1d]

3. Locate the poles and their order for the function $f\left(z\right)=\frac{1}{z(sin\pi z)(z+\frac{1}{2})}.$ Also, find the residue of $f\left(z\right)$ at these poles. [2024/15m/3a]

4. Classify the singular point $z=0$ of the function $f\left(z\right)=\frac{e^z}{z-sinz}$ and obtain the principal part of its Laurent series expansion. [2023/15m/4b]

5. Expand $f\left(z\right)=\frac{1}{{(z-1)}^2(z-3)}$ in a Laurent series valid for the regions (i) $0<|z-1|<2$ and (ii) $0<|z-3|<2$. [2022/10m/1d]

6. Find the Laurent series expansion of $f\left(z\right)=\frac{z^2-z+1}{z(z^2-3z+2)}$ in the powers of $(z+1)$ in the region $|z+1|>3$. [2021/20m/2c]

7. Let $f$ be an entire function whose Taylor series expansion with centre $z=0$ has infinitely many terms. Show that $z=0$ is an essential singularity of $f\left(\frac{1}{z}\right)$. [2021/15m/3a]

8. Show that an isolated singular point $z_0$ of a function $f\left(z\right)$ is a pole of order $m$ if and only if $f\left(z\right)$ can be written in the form $f\left(z\right)=\frac{\varphi \left(z\right)}{{(z-z_0)}^m}$ where $\varphi \left(z\right)$ is analytic and non zero at $z_0$. Moreover $\underset{z=z_0}{Res}f\left(z\right)=\frac{{\varphi }^{(m-1)}\left(z_0\right)}{(m-1)!}$ if $m\ge 1$. [2019/15m/2d]

9. Obtain the first three terms of the Laurent series expansion of the function $f\left(z\right)=\frac{1}{(e^z-1)}$ about the point $z=0$ valid in the region $0<\left|z\right|<2\pi$. [2019/10m/4b]

10. Find the Laurent’s series which represent the function $\frac{1}{(1+z^2)(z+2)}$ when (i) $\left|z\right|<1$ (ii) $1<\left|z\right|<2$ (iii) $\left|z\right|>2$ [2018/15m/4b]

11. Determine all entire functions $f\left(z\right)$ such that 0 is a removable singularity of $f\left(\frac{1}{z}\right)$. [2017/10m/1d]

12. For a function $f:ℂ\to ℂ$ and $n\ge 1$, let $f^{\left(n\right)}$ denote the $n^{\mathrm{\text{th}}}$ derivative of $f$ and $f^{\left(0\right)}=f$. Let $f$ be an entire function such that for some $n\ge 1$, $f^{\left(n\right)}\left(\frac{1}{k}\right)=0$ for all $k=1,2,3,\dots$. Show that $f$ is a polynomial. [2017/15m/4a]

13. Prove that every power series represents an analytic function inside its circle of convergence. [2016/20m/4c]

14. Find all possible Taylor’s and Laurent’s series expansions of the function $f\left(z\right)=\frac{2z-3}{z^2-3z+2}$ about the point $z=0$. [2015/20m/2c]

15. Expand in Laurent series the function $f\left(z\right)=\frac{1}{z^2(z-1)}$ about $z=0$ and $z=1$. [2014/10m/1d]

16. Expand the function $f\left(z\right)=\frac{1}{(z+1)(z+3)}$ in Laurent series valid for (i) $1<\left|z\right|<3$ (ii) $\left|z\right|>3$ (iii) $0<|z+1|<2$ (iv) $\left|z\right|<1$ [2012/15m/3c]

17. Find the Laurent series for the function $f\left(z\right)=\frac{1}{1-z^2},$ with centre $z=1$. [2011/15m/3d]

18. Find the Laurent series of the function $f\left(z\right)=exp\left[\frac{\lambda }{2}(z-\frac{1}{z})\right]\mathrm{\text{as}}{\sum }_{n=-\infty }^{\infty }{C}_n{z}^n\mathrm{\text{for}}0<\left|z\right|<\infty$ where $C_n=\frac{1}{\pi }{\int }_0^{\pi }cos(n\varphi -\lambda sin\varphi )d\varphi ,n=0,\pm 1,\pm 2,\dots$ with $\lambda$ a given complex number and taking the unit circle $C$ given by $z=e^{i\varphi }$ $(-\pi \le \varphi \le \pi )$ as contour in this region. [2010/15m/4b]

3.4] Residues and contour integration

24 questions

1. Use the method of contour integration to prove ${\int }_0^{2\pi }\frac{{sin}^2\theta }{a+bcos\theta }d\theta =\frac{2\pi }{b^2}\{a-\sqrt{a^2-b^2}\},\mathrm{\text{if}}a>b>0.$ [2026/20m/2c]

2. Use the method of contour integration to prove that ${\int }_{-\infty }^{\infty }\frac{x^2-x+2}{x^4+10x^2+9}dx=\frac{5\pi }{12}$. [2025/20m/2c]

3. Evaluate the integral ${\oint }_C^{}\frac{e^z}{z^2{(z+1)}^3}dz$, $C:\left|z\right|=2$. [2025/15m/3a]

4. Locate the poles and their order for the function $f\left(z\right)=\frac{1}{z(sin\pi z)(z+\frac{1}{2})}.$ Also, find the residue of $f\left(z\right)$ at these poles. [2024/15m/3a]

5. Evaluate ${\int }_0^{2\pi }\frac{cos2\theta }{5+4cos\theta }d\theta$ using contour integration. [2023/20m/2c]

6. Apply the calculus of residues to evaluate ${\int }_{-\infty }^{\infty }\frac{cosxdx}{(x^2+a^2)(x^2+b^2)}$, $a>b>0$. [2022/20m/2c]

7. Evaluate ${\int }_C^{}\frac{z+4}{z^2+2z+5}dz$, where $C$ is $|z+1-i|=2$. [2022/15m/3a]

8. Let $c:[0,1]\to ℂ$ be the curve, where $c\left(t\right)=e^{4\pi it}$, $0\le t\le 1$. Evaluate the contour integral ${\int }_c^{}\frac{dz}{2z^2-5z+2}$. [2021/10m/1d]

9. Using contour integration, evaluate the integral ${\int }_{-\infty }^{\infty }\frac{sinxdx}{x(x^2+a^2)}$, $a>0$. [2021/20m/4b]

10. Using contour integration, evaluate the integral ${\int }_0^{2\pi }\frac{1}{3+2sin\theta }d\theta$. [2020/20m/2c]

11. Show that an isolated singular point $z_0$ of a function $f\left(z\right)$ is a pole of order $m$ if and only if $f\left(z\right)$ can be written in the form $f\left(z\right)=\frac{\varphi \left(z\right)}{{(z-z_0)}^m}$ where $\varphi \left(z\right)$ is analytic and non zero at $z_0$. Moreover $\underset{z=z_0}{Res}f\left(z\right)=\frac{{\varphi }^{(m-1)}\left(z_0\right)}{(m-1)!}$ if $m\ge 1$. [2019/15m/2d]

12. Show by applying the residue theorem that ${\int }_0^{\infty }\frac{dx}{{(x^2+a^2)}^2}=\frac{\pi }{4a^3}$, $a>0$. [2018/15m/3b]

13. Using contour integral method, prove that ${\int }_0^{\infty }\frac{xsinmx}{a^2+x^2}dx=\frac{\pi }{2}{e}^{-ma}.$ [2017/15m/2b]

14. Let $\gamma :[0,1]\to ℂ\mathrm{\text{be the curve}}\gamma \left(t\right)=e^{2\pi it},0\le t\le 1.$ Find, giving justifications, the value of the contour integral ${\int }_{\gamma }^{}\frac{dz}{4z^2-1}$ [2016/15m/3c]

15. State Cauchy’s residue theorem. Using it, evaluate the integral ${\int }_C^{}\frac{e^z+1}{z(z+1){(z-i)}^2}dz;C:\left|z\right|=2$ [2015/15m/3a]

16. Evaluate the integral ${\int }_0^{\pi }\frac{d\theta }{{(1+\frac{1}{2}cos\theta )}^2}$ using residues. [2014/20m/3c]

17. Prove that if $b{e}^{a+1}<1$ where $a$ and $b$ are positive and real, then the function $z^n{e}^{-a}-b{e}^z$ has $n$ zeroes in the unit circle. [2013/10m/1d]

18. Using Cauchy’s residue theorem, evaluate the integral $I={\int }_0^{\pi }{sin}^4\theta d\theta$ [2013/15m/4b]

19. Evaluate by contour integration $I={\int }_0^{2\pi }\frac{d\theta }{1-2acos\theta +a^2},a^2<1.$ [2012/15m/3d]

20. Evaluate by Contour integration, ${\int }_0^1\frac{dx}{{(x^2-x^3)}^{1/3}}$ [2011/15m/3c]

21. Let $f\left(z\right)=\frac{a_0+a_{1}z+\cdots \cdots +a_{n-1}{z}^{n-1}}{b_0+b_{1}z+\cdots \cdots +b_n{z}^n}$, $b_n\ne 0$. Assume that the zeroes of the denominator are simple. Show that the sum of the residues of $f\left(z\right)$ at its poles is equal to $\frac{a_{n-1}}{b_n}$. [2009/12m/1e]

22. If $\alpha$, $\beta$, $\gamma$ are real numbers such that ${\alpha }^2>{\beta }^2+{\gamma }^2$ show that : ${\int }_0^{2\pi }\frac{d\theta }{\alpha +\beta cos\theta +\gamma sin\theta }=\frac{2\pi }{\sqrt{{\alpha }^2-{\beta }^2-{\gamma }^2}}.$ [2009/30m/4a]

23. Find the residue of $\frac{cotzcothz}{z^3}\mathrm{\text{at}}z=0$ [2008/12m/1e]

24. Evaluate ${\int }_C^{}[\frac{e^z}{z^2(z^2+2z+2)}+log(z-6)+\frac{1}{{(z-4)}^2}]dz$ where $C$ is the circle $\left|z\right|=3$. State the theorems you use in evaluating above integral. [2008/15m/4a]

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Last updated: 30 October 2026