Table of Contents

**MTH632: Complex Analysis and Differential Geometry**

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra, and multilinear algebra. MTH632 Handouts pdf

# MTH632 Handouts pdf

**Course Category: mathematics **

**Course Outline**

Representation of complex numbers, Algebraic properties of complex numbers, Geometric representation of complex numbers, Complex conjugates, Exponential forms, Algebraic properties of exponential form, DeMoiver’s Theorem, Regions on the complex plane, Functions of a complex variable, Mappings, Limits, Theorems on limits, Limits involving the point at infinity, Continuity, Derivative, Differentiation formulas, Cauchy-Reimann Equations, Cauchy-Reimann Equations in Polar coordinates, Analytic functions, Harmonic functions, Uniquely determined analytic functions, Reflection Principle, Exponential function, Logarithmic functions, Branches and derivatives of Logarithms, Identities involving Logarithms, Complex exponents, Trigonometric functions,

Hyperbolic functions, Inverse trigonometric and hyperbolic functions, Inverse trigonometric and hyperbolic functions, Derivative of functions, Definite integrals of functions, Contours, Contour integrals, Upper bounds for moduli of contour integrals, Antiderivatives, Cauchy-Goursat theorem, Domains, Cauchy Integral Formula, Liouville’s theorem, Sequence of complex numbers, Convergence of sequence, Series, Taylor Series, Laurent Series, Continuity and power series, Differentiation of power series, Integration of power series, uniqueness of series representation, Algebra of power series, Isolated singular points, Residues, Cauchy Residue theorem, Residues at infinity, Isolated singular points, Residues at poles, Zeros of an analytic function, Zeros and poles, Functions and isolated singular points. MTH632 Handouts pdf

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### MTH632 HANDOUTS

**MTH632: Complex Analysis and Differential Geometry munity**

It is an extremely useful and beautiful part of mathematics and forms the basis of many techniques used in many branches of mathematics and physics. We will extend the concepts of derivatives and integrals, known from calculus to the case of complex functions of a complex variable. In doing so, we will encounter the analytical functions that form the core of this part of the course. In fact, complex analysis is largely a study of analytical functions. pp.

#### Polar form and argument function

Points in a plane can also be represented using polar coordinates, and this representation, in turn, translates into a complex number representation. Let (x, y) be a point in the plane. If we define r = θ r ✼ z = reiθ p x 2 + y 2 and θ by θ = arctan(y/x), then we can write (x, y) = (r cos θ, r sin θ) = r ( cos θ ,sin θ). The complex number z = x + i y can then be written as z = r (cos θ + i sin θ). The real number r, as we have seen, is modulo |z| z, and the complex number cos θ + i sin θ has unit modulus.

Comparing the Taylor series for the cosine and sine functions and the exponential function, we note that cos θ+isin θ = e iθ. The angle θ is called the z argument and is written arg(z).