Unit 4 of 10
Complex analysis
Six or seven questions, mostly residues, contour integrals and Möbius maps.
- 6–7questions expected next time
- 6asked in 2025
- 7asked in 2026
- 2ideas asked in both papers
Residues are the core skill. Most contour questions are one residue away.
Most likely questions
Practise each one until you can do it without notes, then tick it. Ticks stay in this browser.
Already repeated
These came up in 2025 and again in 2026. Answer each in your head, then tap to check.
Möbius map: image of a circle or the real axis
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Circles and lines go to circles and lines. Three points and their images fix the map.
∫₀^2π dθ / (a + b sin θ) by residues
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2π / √(a² − b²), for a > |b|.
Asked in 2026
7 questions. Tap one for the answer and working.
Q70F analytic and conformal at z0 implies f'(z0) nonzeroA
Analytic and conformal at z0 iff f'(z0) != 0
Sure
Q77Number of zeros of z^5-2z+1 (OCR-uncertain) insideA
z^5-2z+1=(z-1)(z^4+z^3+z^2+z-1). Roots (computed): 1, 0.5188, -1.2906, -0.114+-1.217i. Only 0.5188 is strictly inside |z|<1 (z=1 is on the circle). Rouche with -2z also gives 1
Sure
Q81W=(z-i)/(z+i)-type map: image of real axisA
z real: |z-i|=|z+i| so |w|=1: image is the unit circle
Sure
Q86F analytic,A
f(z)/z^3 is entire and bounded by 1 (f has zero of order >=3 at 0) -> constant by Liouville: f=a z^3, |a|<=1
Sure
Q92Integral over [0,2pi] of 1/(2+sin theta) d theta (OCR-uncertain)A
Integrand = d/dtheta ln(2+sin theta); over full period integral = 0
Sure
Q95F analytic in simply connected domain, integral over every closed curve is 0, then f is ...D
Integrand is conj(f(z)) (bar visible in image). Green: closed integral of conj(f) dz = 2i * Double integral conj(f') dA = 0 for all C -> f'=0 -> f constant. (If bar were absent, answer would be trivially "analytic")
Sure
The overbar on f(z) is printed in both English and Hindi. Without it, every analytic f would work.
Q100F=u+iv entire with u^2+v^2 = exp(2x+...) (OCR-uncertain) and f(0)=1: find fLikely bonus marksD
Option D is NOT visible in the photo (cut off). Math: |f'|^2 = (1/4)Laplacian(e^2x+y^2) = e^2x+1/2, but log|f'|^2 would have to be harmonic (f' zero-free), and it is not -> no entire f exists. A, B, C all fail (|e^z|^2=e^2x; |e^z+iz|^2 and |e^z+z|^2 have extra cross terms). Guess D = "no such function"
Not sure
Option D is cut off in the photo. No entire function satisfies the condition and A–C all fail, so D is right only if it reads "none of these". The second solve guessed A.
Asked in 2025
6 questions. Tap one for the answer and working.
Q61Residue at z=0C
z cos(1/z) = z - 1/(2z) + ...; residue -1/2.
Matches the official key
Q62Image ofA
z=(2w+5)/(4(w+1)); |z|=1 => u^2+v^2+u-3/4=0, centre (-1/2,0), radius 1.
Matches the official key
Q63∮ overB
Residue at z=-1: (1/3!) d^3/dz^3 e^{2z} = 8e^{-2}/6; integral = 2 pi i * 8/(6e^2) = 8 pi i/(3e^2).
Matches the official key
Q64F(z) given u = eˣ(x cos y − y sin y)A
f = z e^z has real part e^x(x cos y - y sin y).
Matches the official key
Q65Value of a definite integral (contour method)B
Integral of d(theta)/(a+b sin theta) over 0..2pi = 2 pi/sqrt(a^2-b^2) = 2 pi/3.
Matches the official key
Q66P/q has pole of order m when q has zero of order mA
q has zero of order m at z0 and p(z0)!=0, so p/q has a pole of order m; R explains A
Matches the official key
In the syllabus, not asked yet
Give each of these one revision pass. Skipped topics often turn up later.
- Liouville's theorem (asked in 2019)
- Taylor and Laurent expansions
- Classifying singularities
Full syllabus for this unit
Analytic functions, Cauchy–Riemann, Cauchy, Morera, Liouville, integral formula, zeros, Taylor and Laurent series, residues, contour integration, conformal maps, Möbius maps.