CO1: Understand the significance of differentiability of complex functions leading to the understanding of Cauchy-Riemann equations.
CO2: Evaluate the contour integrals and understand the role of Cauchy-Goursat theorem and the Cauchy integral formula.
CO3: Expand some simple functions as their Taylor and Laurent series, get familiar with the linear transformation and Mobius transformation.
CO4: Analyze sequences and series of complex numbers and functions, uniform convergence, power series, Taylor and Laurent series expansions, and classify isolated singularities.
CO5: Master conformal mappings, the Jacobian of transformations, elementary transformations (translation, rotation, magnification, inversion), Möbius (bilinear) transformations, cross-ratio preservation, and fixed points.
📝Module:Unit 1 — Introduction to Complex Numbers, Analytic Functions, and Cauchy-Riemann Equations | 8
Introduction to complex numbers and their geometrical interpretation in the Argand plane, algebraic operations and properties of $\mathbb{C}$, Cartesian and polar representations ($z = x + iy = r e^{i\theta}$), modulus $|z|$, conjugate $\bar{z}$, triangle inequalities ($||z_1| - |z_2|| \le |z_1 \pm z_2| \le |z_1| + |z_2|$), De Moivre's theorem, and $n$-th roots of unity. Euclidean topology of the complex plane: open and closed disks, neighborhoods $B_\epsilon(z_0)$, interior, boundary, and limit points, open and closed sets, connectedness, domains, regions, and extended complex plane $\mathbb{C}_\infty$ via stereographic projection onto the Riemann sphere. Functions of a complex variable, mapping definition $w = f(z) = u(x,y) + i v(x,y)$, domain and range, geometric visualization. Limits of complex functions, formal $\epsilon$-$\delta$ definition, equivalence with real component limits, path-independence across infinite directions of approach, and limits involving infinity. Continuity of complex functions, continuity on domains, algebra and composition of continuous functions, and uniform continuity on compact sets (Heine-Cantor theorem). Complex differentiability: derivative $f'(z_0)$ as a limit of difference quotients, linear approximation, contrast with real differentiability in $\mathbb{R}^2$, non-differentiability of $\bar{z}$, $|z|^2$, and $\operatorname{Re}(z)$. Cauchy-Riemann equations in Cartesian form: necessary conditions ($u_x = v_y$ and $u_y = -v_x$) derived from orthogonal paths, derivative formula $f'(z) = u_x + i v_x = v_y - i u_y$. Sufficient conditions for complex differentiability ($C^1$ condition on partial derivatives). Polar form of Cauchy-Riemann equations ($u_r = \frac{1}{r}v_\theta$ and $v_r = -\frac{1}{r}u_\theta$) and derivative $f'(z) = e^{-i\theta}(u_r + i v_r)$. Analytic (holomorphic) and regular functions on open domains, entire functions, singularities, harmonic functions satisfying Laplace's equation $\nabla^2 u = 0$, harmonic conjugates $v(x,y)$, orthogonal families of level curves, and Milne-Thomson method for analytic function reconstruction.
The complex exponential function: definition $e^z = e^x(\cos y + i\sin y)$, fundamental properties ($e^{z_1+z_2} = e^{z_1}e^{z_2}$, $|e^z| = e^x$, $\arg(e^z) = y + 2k\pi$), periodicity with fundamental period $2\pi i$, mapping of Cartesian coordinate lines, complex derivative $\frac{d}{dz}e^z = e^z$, and non-vanishing property. Complex trigonometric and hyperbolic functions: definitions of $\sin z$, $\cos z$, $\sinh z$, $\cosh z$, real and imaginary decompositions, zeroes, unboundedness of $|\sin z|$ and $|\cos z|$ in $\mathbb{C}$, fundamental identities and derivatives. The complex logarithmic function: multi-valued logarithm $\log z = \ln|z| + i\arg z = \ln|z| + i(\operatorname{Arg} z + 2k\pi)$ ($z \ne 0$), principal branch $\operatorname{Log} z = \ln|z| + i\operatorname{Arg} z$ ($-\pi < \operatorname{Arg} z \le \pi$), algebraic properties, and failure of real logarithmic identities. Branches and branch cuts of the logarithm: branch points at $z = 0$ and $z = \infty$, analyticity of the principal branch $\operatorname{Log} z$ on the cut plane $\mathbb{C} \setminus (-\infty, 0]$, derivative $\frac{d}{dz}\operatorname{Log} z = 1/z$, general branch cuts along arbitrary rays. Complex exponents and inverse trigonometric functions: general complex powers $z^c = e^{c\log z}$, principal value P.V. $z^c = e^{c\operatorname{Log} z}$, multi-valuedness, inverse functions $\arcsin z$, $\arccos z$, $\arctan z$, and logarithmic formulas. Analyticity and derivatives of elementary functions, chain rule compositions, and singular sets. Definite integrals of complex-valued functions of a real variable: $\int_a^b w(t)\,dt = \int_a^b u(t)\,dt + i\int_a^b v(t)\,dt$, linearity, Fundamental Theorem of Calculus, and modulus inequality $|\int_a^b w(t)\,dt| \le \int_a^b |w(t)|\,dt$. Contours and smooth paths in the complex plane: parameterized paths $z(t) = x(t) + iy(t)$ ($a \le t \le b$), smooth arcs, piecewise smooth contours, simple closed curves, Jordan curve theorem, orientation, reverse paths $-C$, and path concatenation. Contour integrals of complex functions: definition $\int_C f(z)\,dz = \int_a^b f(z(t))z'(t)\,dt$, independence of orientation-preserving reparameterization, linearity, decomposition into real line integrals $\int_C (u\,dx - v\,dy) + i\int_C (v\,dx + u\,dy)$, integration along line segments, circular arcs, and polygonal paths. Upper bounds for moduli of contour integrals: rigorous statement and proof of the $ML$-inequality ($|\int_C f(z)\,dz| \le M L$), applications to circular contours, and asymptotic vanishing bounds as $R \to \infty$.
📝Module:Unit 3 — The Cauchy-Goursat Theorem, Integral Formulas, and Classical Theorems | 8
Complex integration and Cauchy's theorem: statement and classical proof of $\oint_C f(z)\,dz = 0$ for simple closed contours assuming continuous derivative $f'(z)$ ($C^1$) via Green's Theorem and Cauchy-Riemann equations. Eliminating the continuous derivative hypothesis: Goursat's lemma for rectangles and triangles via quaternary nested bisection, diameter decay, and Taylor increment bounding proving $\oint_{\partial R} f(z)\,dz = 0$. The Cauchy-Goursat theorem on simply connected domains: extension to arbitrary simple closed contours, topological invariance, and path independence. Primitives and the fundamental theorem of complex integration: existence of single-valued antiderivative $F(z)$ such that $F'(z) = f(z)$, equivalence with path independence, and evaluation formula $\int_{z_1}^{z_2} f(z)\,dz = F(z_2) - F(z_1)$. Cauchy-Goursat theorem for multiply connected domains: deformation of contours principle, cross-cuts and keyhole channels, equality $\oint_{C_1} f(z)\,dz = \oint_{C_2} f(z)\,dz$ for nested contours enclosing isolated singularities. Cauchy's Integral Formula (CIF): fundamental representation $f(z_0) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z - z_0}\,dz$ for interior points $z_0$, complete proof via contour deformation to an $\epsilon$-circle and continuity of $f$, Gauss's mean value theorem for analytic functions. Cauchy's Integral Formula for higher derivatives: $f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_C \frac{f(z)}{(z - z_0)^{n+1}}\,dz$, deduction that every analytic function is infinitely differentiable ($f \in C^\infty$) with all derivatives analytic. Morera's theorem: complete statement and proof of the converse of Cauchy's theorem for continuous functions with vanishing loop integrals. Cauchy's inequality for derivatives: $|f^{(n)}(z_0)| \le \frac{n! M_R}{R^n}$. Liouville's theorem: every bounded entire function is constant (proof via $n=1$ Cauchy estimate as $R \to \infty$). The Fundamental Theorem of Algebra: rigorous proof via Liouville's theorem applied to $1/P(z)$. Poisson's Integral Formula for a circle: derivation from Cauchy's Integral Formula via symmetric points $z^* = R^2/\bar{z}$ and real part extraction, Poisson kernel $P(R, r, \theta - \phi)$, and harmonic boundary-value geometry.
📝Module:Unit 4 — Complex Sequences, Series, Taylor Expansions, and Laurent Expansions | 8
Convergence of complex sequences: definition $z_n \to z_0$, equivalence with component sequences $x_n \to x_0$ and $y_n \to y_0$, Cauchy criterion for convergence, boundedness of convergent sequences. Infinite series of complex numbers: partial sums $S_N = \sum_{n=1}^N z_n$, convergence, absolute convergence implying convergence, divergence test ($\lim z_n \ne 0$), geometric series $\sum_{n=0}^\infty z^n = \frac{1}{1 - z}$ for $|z| < 1$. Sequences and series of functions: pointwise vs. uniform convergence on sets in $\mathbb{C}$, Cauchy criterion for uniform convergence, Weierstrass $M$-test for complex series. Properties of uniformly convergent series: continuity of the sum function, term-by-term contour integration $\int_C (\sum f_n(z))\,dz = \sum \int_C f_n(z)\,dz$, term-by-term differentiation of series of analytic functions. Power series and radius of convergence: power series $\sum_{n=0}^\infty a_n(z - z_0)^n$, Cauchy-Hadamard formula $R = 1/\limsup |a_n|^{1/n}$, ratio test formula $R = \lim |a_n/a_{n+1}|$, circle and disk of convergence $|z - z_0| < R$. Analyticity and term-by-term operations: analyticity of the sum function inside the open disk of convergence, identical radius of convergence under term-by-term differentiation and integration. Taylor's theorem in the complex plane: statement and rigorous proof that any function analytic in $|z - z_0| < R_0$ has unique expansion $f(z) = \sum_{n=0}^\infty \frac{f^{(n)}(z_0)}{n!}(z - z_0)^n$ using Cauchy's Integral Formula and uniform convergence of geometric series. Standard Taylor series expansions for $e^z$, $\sin z$, $\cos z$, $\sinh z$, $\cosh z$, $\frac{1}{1-z}$, $\operatorname{Log}(1+z)$, and nearest singularity rule for radius of convergence. Laurent's theorem and annulus of convergence: expansion of function analytic in open circular annulus $R_1 < |z - z_0| < R_2$ into two-sided series $f(z) = \sum_{n=-\infty}^\infty c_n(z - z_0)^n = \sum_{n=0}^\infty a_n(z - z_0)^n + \sum_{n=1}^\infty \frac{b_n}{(z - z_0)^n}$, rigorous proof via CIF on concentric circles with cross-cut keyholes. Uniqueness of Taylor and Laurent series representations. Classification of isolated singularities via Laurent series principal parts: removable singularities (no negative powers; Riemann's removable singularity theorem), poles of order $m$ (finite negative powers), and essential singularities (infinitely many negative powers; Casorati-Weierstrass theorem).
Linear transformations: affine mapping $w = Az + B$ ($A \ne 0$), geometric decomposition into magnification $|A|$, rotation by $\arg A$, and translation by $B$, preservation of Euclidean angles, straight lines, and circles. The Jacobian of a transformation: mapping of coordinate planes $(x,y) \mapsto (u,v)$, derivation of Jacobian determinant $J = u_x v_y - u_y v_x = u_x^2 + v_x^2 = |f'(z)|^2$ via Cauchy-Riemann equations, non-singularity condition $f'(z) \ne 0$. Conformal mappings and angle preservation: definition of conformality (preservation of angle magnitude and sense of rotation between intersecting smooth curves), proof that an analytic function $f(z)$ is conformal at $z_0$ if and only if $f'(z_0) \ne 0$, scale factor $|f'(z_0)|$, rotation angle $\arg f'(z_0)$, critical points where $f'(z_0) = 0$ (magnification of angles). Elementary transformations: translation $w = z + c$, pure rotation $w = e^{i\alpha} z$, magnification/dilation $w = k z$ ($k > 0$), mapping geometry of half-planes, strips, and sectors. Inversion transformation: geometric inversion $w = 1/z$, reflection across the unit circle followed by complex conjugation, polar action $r e^{i\theta} \mapsto \frac{1}{r}e^{-i\theta}$, mapping of the family of generalized circles and lines $A(x^2 + y^2) + Bx + Cy + D = 0$ into generalized circles and lines. Möbius transformations (bilinear / fractional linear transformations): definition $w = T(z) = \frac{az + b}{cz + d}$ with $ad - bc \ne 0$, bijective mapping on the extended complex plane $\mathbb{C}_\infty$ ($T(-d/c) = \infty$, $T(\infty) = a/c$), group property under composition, and matrix representation in $\mathrm{PGL}_2(\mathbb{C})$. Decomposition theorem: decomposition of any non-trivial Möbius transformation into a finite composition of translations, inversions, and magnifications/rotations, corollary that Möbius maps preserve the family of generalized circles. Cross-ratio of four points: definition $(z_1, z_2, z_3, z_4) = \frac{(z_1 - z_2)(z_3 - z_4)}{(z_1 - z_4)(z_3 - z_2)}$, conventions with $\infty$. Invariance of cross-ratio under Möbius transformations: $(T(z_1), T(z_2), T(z_3), T(z_4)) = (z_1, z_2, z_3, z_4)$, determination of unique transformation mapping three distinct points $(z_1, z_2, z_3)$ to $(w_1, w_2, w_3)$ via implicit cross-ratio equation $(w, w_1, w_2, w_3) = (z, z_1, z_2, z_3)$. Fixed points of a bilinear transformation: roots of $cz^2 + (d - a)z - b = 0$, classification into identity, parabolic (one fixed point), and two distinct fixed points. Canonical conformal mappings between standard domains, including mapping the upper half-plane $\mathbb{H}$ onto the open unit disk $\mathbb{D}$ via $w = e^{i\theta_0}\frac{z - z_0}{z - \bar{z}_0}$ ($\operatorname{Im}(z_0) > 0$).
📝Textbooks
Brown, James Ward & Churchill, R. V. (2014). Complex Variables and Applications (9th ed.). McGraw-Hill Education. New York.
Ponnusamy, S. (2011). Foundations of Complex Analysis (2nd ed.). Alpha Science International Ltd. UK / Narosa Publishing House, New Delhi.
Bak, Joseph & Newman, Donald J. (2010). Complex Analysis (3rd ed.). Undergraduate Texts in Mathematics, Springer. New York.
Zill, Dennis G. & Shanahan, Patrick D. (2003). A First Course in Complex Analysis with Applications. Jones & Bartlett Publishers, Inc.
Mathews, John H. & Howell, Russell W. (2012). Complex Analysis for Mathematics and Engineering (6th ed.). Jones & Bartlett Learning. Narosa, Delhi.
02Complex Numbers, Analytic Functions, and Cauchy-Riemann EquationsCH
03Elementary Complex Functions, Branches, and Contour IntegralsCH
04The Cauchy-Goursat Theorem, Integral Formulas, and Classical TheoremsCH
05Complex Sequences, Series, Taylor, and Laurent ExpansionsCH
06Conformal Mappings, Elementary Transformations, and Möbius MapsCH