Complex Analysis
92 questions across 6 topics. Also examined in CNET.
Algebra of Complex Numbers, Power Series, Transcendental Functions
- 21Complex analysis — modulus of an exponentialMSET June 2025Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 22Complex analysis — stereographic projectionMSET June 2025Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 28Complex analysis — radii of convergenceMSET June 2025Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 32Complex analysis — logarithms of non-vanishing functionsMSET June 2025Algebra of Complex Numbers, Power Series, Transcendental FunctionsHard
- 34Complex analysis — a square root of 1+iMSET June 2025Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 23Roots of unity — which power sum failsMSET April 2024Algebra of Complex Numbers, Power Series, Transcendental FunctionsHard
- 52Homeomorphism type of plane curves — compact loops versus unbounded arcsMSET April 2024Algebra of Complex Numbers, Power Series, Transcendental FunctionsMedium
- 21Evaluating a complex power directly from the principal logarithmMSET March 2023Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 23Radius of convergence of a geometric-type power seriesMSET March 2023Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 24The family of disk automorphisms sending 1/2 to 0MSET March 2023Algebra of Complex Numbers, Power Series, Transcendental FunctionsMedium
- 27Image of a square under the exponential mapMSET March 2023Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 35Degree of the splitting field of x^3-2MSET March 2023Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 3A universal inequality between x+y and zMSET September 2021Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 4Complex cosine — unbounded, not injective, yet surjectiveMSET September 2021Algebra of Complex Numbers, Power Series, Transcendental FunctionsMedium
- 43The complex-squaring map is an open map away from its critical pointMSET September 2021Algebra of Complex Numbers, Power Series, Transcendental FunctionsMedium
- 49Counting irreducible quadratic factors of X^8+1MSET September 2021Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 3The two-branch characterization of zw, z+winRMSET December 2020Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 4Re-centering the geometric series for 1/z at z=1MSET December 2020Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 16Closer to i than to -i means lying in the upper half planeMSET December 2020Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 17The strip 0<Re(z)lepi/2 maps onto the right half plane under sin zMSET December 2020Algebra of Complex Numbers, Power Series, Transcendental FunctionsMedium
- 47Counting maximal ideals via the irreducible factorization of X^8+1 over RMSET December 2020Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 4Evaluating i^i via the principal logarithmMSET June 2019Algebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 46Choosing g=-f+c makes bar f+bar g collapse to a constantMSET June 2019Algebra of Complex Numbers, Power Series, Transcendental FunctionsHard
- 31/(-1+i)=(-1-i)/2 lies in the third quadrantMSET January 2018 Paper IIAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 27sin(iy)=isinh y and cos(iy)=cosh yMSET January 2018 Paper IIAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 28Collinear points have proportional difference vectorsMSET January 2018 Paper IIAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 34The product of the mth roots of unity is (-1)^m+1MSET January 2018 Paper IIAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 15Ratio test: the 2^n in the denominator sets the radius to 2MSET January 2018 Paper IIIAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 31/(3+3i)=(1-i)/6MSET April 2017 Paper IIAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 26The exponential series converges everywhereMSET April 2017 Paper IIAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 15Gap series sum z^n^2: coefficients 0 or 1, so R=1MSET April 2017 Paper IIIAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
Taylor Series, Laurent Series, Calculus of Residues
- 24Complex analysis — a removable singularityMSET June 2025Taylor Series, Laurent Series, Calculus of ResiduesEasy
- 33Complex analysis — Casorati–WeierstrassMSET June 2025Taylor Series, Laurent Series, Calculus of ResiduesMedium
- 30Injectivity on a small punctured disk — e^z versus e^1/zMSET April 2024Taylor Series, Laurent Series, Calculus of ResiduesHard
- 32Residue at a third-order poleMSET April 2024Taylor Series, Laurent Series, Calculus of ResiduesMedium
- 33Residue from the Laurent series of (e^z-1)/z^2MSET April 2024Taylor Series, Laurent Series, Calculus of ResiduesEasy
- 34Inversion of a circle through the poleMSET April 2024Taylor Series, Laurent Series, Calculus of ResiduesEasy
- 2The alternating harmonic seriesMSET March 2023Taylor Series, Laurent Series, Calculus of ResiduesEasy
- 28Residue extraction from the Laurent series of z^4sin(1/z)MSET March 2023Taylor Series, Laurent Series, Calculus of ResiduesEasy
- 33Residue of cot z at a pole on the real axisMSET March 2023Taylor Series, Laurent Series, Calculus of ResiduesEasy
- 19Contour integrals around concentric circles agree, by deformationMSET September 2021Taylor Series, Laurent Series, Calculus of ResiduesMedium
- 19Extracting a residue from the Laurent series of cos(1/z)MSET December 2020Taylor Series, Laurent Series, Calculus of ResiduesEasy
- 20A third-order pole hiding behind an apparent double zeroMSET December 2020Taylor Series, Laurent Series, Calculus of ResiduesEasy
- 69Spotting 13^2equiv40pmod43 among the given optionsMSET December 2020Taylor Series, Laurent Series, Calculus of ResiduesEasy
- 19A doubly-wound contour picks up twice the residueMSET June 2019Taylor Series, Laurent Series, Calculus of ResiduesEasy
- 20Poles of csc(1/(z-1)) accumulate at z=1 itselfMSET June 2019Taylor Series, Laurent Series, Calculus of ResiduesEasy
- 4Residues pmtfrac12 cancel on a contour around both poles — but no contour is printedMSET January 2018 Paper IITaylor Series, Laurent Series, Calculus of ResiduesEasy
- 18A geometric series in -z^2MSET January 2018 Paper IITaylor Series, Laurent Series, Calculus of ResiduesEasy
- 30Only the rational function has a pole; the others are essential or entireMSET January 2018 Paper IITaylor Series, Laurent Series, Calculus of ResiduesEasy
- 16Casorati–Weierstrass: only an essential singularity comes arbitrarily close to every valueMSET January 2018 Paper IIITaylor Series, Laurent Series, Calculus of ResiduesMedium
- 19sec z has a pole at every zero of cos zMSET January 2018 Paper IIITaylor Series, Laurent Series, Calculus of ResiduesEasy
- 18The Maclaurin series of cos zMSET April 2017 Paper IITaylor Series, Laurent Series, Calculus of ResiduesEasy
- 31e^1/(z-2) has an essential singularity at z=2; z=1 is only a poleMSET April 2017 Paper IITaylor Series, Laurent Series, Calculus of ResiduesEasy
- 16Only an essential singularity lets f(zn) have no limit, even inftyMSET April 2017 Paper IIITaylor Series, Laurent Series, Calculus of ResiduesMedium
Liouville's Theorem, Maximum Modulus Principle, Open Mapping Theorem
- 25Complex analysis — Schwarz's lemmaMSET June 2025Liouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 29Complex analysis — when an analytic function is constantMSET June 2025Liouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 31Complex analysis — zeros of entire and analytic functionsMSET June 2025Liouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 29Injective entire functions — forced to be linearMSET April 2024Liouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 32Two theorems in tandem — a bounded-real-part trick and equality in Schwarz–PickMSET March 2023Liouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremHard
- 34Casorati–Weierstrass — the image near an essential singularity is denseMSET March 2023Liouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 44Picard's theorem — what a non-constant entire function's range can look likeMSET September 2021Liouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 44Only linear polynomials give bijective entire functionsMSET June 2019Liouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremEasy
- 29Liouville's theoremMSET January 2018 Paper IILiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremEasy
- 17z takes only real values, so its image of an open set is never open in CMSET January 2018 Paper IIILiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremEasy
- 32Zeros can accumulate at the boundary without fequiv0MSET April 2017 Paper IILiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 171/z is analytic and non-constant on Csetminus0, hence openMSET April 2017 Paper IIILiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremEasy
- 19Maximum modulus principleMSET April 2017 Paper IIILiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremEasy
Conformal Mappings, Mobius Transformations
- 27Complex analysis — disk to right half-planeMSET June 2025Conformal Mappings, Mobius TransformationsEasy
- 30Complex analysis — conformality on the unit diskMSET June 2025Conformal Mappings, Mobius TransformationsMedium
- 25Recognising an automorphism of the unit diskMSET April 2024Conformal Mappings, Mobius TransformationsHard
- 25A Möbius map with infty as its only fixed point is a translationMSET March 2023Conformal Mappings, Mobius TransformationsEasy
- 16The standard Möbius map from the unit disc to the right half-planeMSET September 2021Conformal Mappings, Mobius TransformationsEasy
- 70One of any four consecutive integers is divisible by 4, killing the Mobius productMSET December 2020Conformal Mappings, Mobius TransformationsEasy
- 16Three points determine a Mobius map, and there are infinitely many valid choicesMSET June 2019Conformal Mappings, Mobius TransformationsEasy
- 42Squaring a complex number is injective away from antipodal pairsMSET June 2019Conformal Mappings, Mobius TransformationsMedium
- 18A M\"obius map omits a/c, its value at infinityMSET January 2018 Paper IIIConformal Mappings, Mobius TransformationsEasy
- 29Four points are concyclic or collinear iff their cross ratio is realMSET April 2017 Paper IIConformal Mappings, Mobius TransformationsMedium
- 18cz^2+(d-a)z-b=0 has at most two rootsMSET April 2017 Paper IIIConformal Mappings, Mobius TransformationsEasy
Contour Integral, Cauchy's Theorem, Cauchy's Integral Formula
- 26Complex analysis — Cauchy's integral formulaMSET June 2025Contour Integral, Cauchy's Theorem, Cauchy's Integral FormulaEasy
- 21Complex analysis — where 1/z has a primitiveMSET April 2024Contour Integral, Cauchy's Theorem, Cauchy's Integral FormulaHard
- 31Argument principle — integrating P'/P around all the rootsMSET April 2024Contour Integral, Cauchy's Theorem, Cauchy's Integral FormulaMedium
- 31Morera's theorem and its converse (for simply connected domains)MSET January 2018 Paper IIContour Integral, Cauchy's Theorem, Cauchy's Integral FormulaMedium
- 32sin z/z has a removable singularity at 0MSET January 2018 Paper IIContour Integral, Cauchy's Theorem, Cauchy's Integral FormulaEasy
- 4Only the 1/z term contributes: residue 1MSET April 2017 Paper IIContour Integral, Cauchy's Theorem, Cauchy's Integral FormulaEasy
- 30Analytic functions are infinitely differentiableMSET April 2017 Paper IIContour Integral, Cauchy's Theorem, Cauchy's Integral FormulaEasy
Analytic Functions, Cauchy-Riemann Equations
- 23Complex analysis — harmonic parts and reflectionMSET June 2025Analytic Functions, Cauchy-Riemann EquationsMedium
- 24Harmonic conjugates — mutual conjugacy forces constantsMSET April 2024Analytic Functions, Cauchy-Riemann EquationsHard
- 22Recovering u from v via the Cauchy–Riemann equationsMSET March 2023Analytic Functions, Cauchy-Riemann EquationsEasy
- 18Harmonic combinations of u,v — why u^2 fails but uv survivesMSET September 2021Analytic Functions, Cauchy-Riemann EquationsMedium
- 18A Liouville-type theorem for non-negative harmonic functions, and logz's missing conjugateMSET December 2020Analytic Functions, Cauchy-Riemann EquationsMedium
- 27A real-valued function satisfies Cauchy–Riemann only where its gradient vanishesMSET April 2017 Paper IIAnalytic Functions, Cauchy-Riemann EquationsEasy
- 28-e^xcos y+1=Re(1-e^z), so its conjugate is -e^xsin yMSET April 2017 Paper IIAnalytic Functions, Cauchy-Riemann EquationsEasy