Table of Contents
1. Review of foundational concepts
1.1. Sequences and Series
1.1.1. Sequences of Real Numbers and their Series – sequences, limits, series, convergence, harmonic numbers, summation by parts, change in the order of summation
1.1.2. Power Series and Generating Functions – definitions, radius of convergence, generating function representations of sequences, convolution
1.2. Integrals
1.2.1. Riemann Sums – definition, direct evaluation of certain sums
1.2.2. Fundamental Theorem - definition of definite integral, statement of theorem, verifications
1.2.3. Multiple Integrals – double integrals, conditions for reversal or order of integration
1.3. Evaluation Techniques
1.3.1. Integration by Parts - review
1.3.2. Conversion to Multiple Integrals – “Feynman’s Technique,” replacing a portion of an integrand with an integral representation and reversing the order of integration
1.3.3. Green’s Theorem – review, path integrals and parametrization, Stokes’ Theorem, applications
1.3.4. Partial Fractions review
1.4. Problems
2. Complex Integration
2.1. Analytic Functions
2.1.1. Cauchy-Riemann Conditions – complex functions and their derivatives, defining analytic functions as a direction-independent derivative, harmonic functions
2.1.2. Evaluating Complex Integrals – numerical examples of parametrizations
2.1.3. Path Independence – demonstrate for analytic functions and demonstrate invalidity for nonanalytic integrands
2.2. Cauchy’s Theorems
2.2.1. Winding Numbers – definition in terms of a complex integral
2.2.2. Cauchy’s Integral Theorem – derivation and illustration for a wide variety of integrands and contours
2.2.3. Cauchy’s Theorem – statement, examples, Liouville’s Theorem, Morera’s Theorem
2.3. Useful Results
2.3.1. Taylor Series – review, error analysis in complex plane, convergence
2.3.2. Laurent Series – regions of validity (e.g., annuli), analytic continuation
2.3.3. Argument Principle – derivation for zeroes and poles
2.3.4. Rouche’s Theorem – derivation, illustration for determining poles within integration contours
2.4. Multivalued Functions – branch points, branch cuts, Riemann surfaces
2.5. Problems
3. Evaluation of Real Integrals and Sums
3.1. Preliminary Matters
3.1.1. Poles and Residue Theory – residue definition, residue computation
3.1.2. Essential Singularities – computation of residues of essential singularities
3.1.3. Branch Points – illustration of a unified approach to expressing an integral of a function in terms of its singularities
3.2. Definite Integrals
3.2.1. Integrands Having Both Poles and Branch Points – e.g., integrands featuring logs and exponents less than -1
3.2.2. Integrands Defined Over - insertion of one higher power of log(z) in the integrand, residue backpropagation
3.2.3. Integrands Having Rational Functions of Polynomials and Trigonometric Functions – integration over the unit circle, modifying the unit circle in the presence of singularities, replacing monomial with a branch point in constructing a contour integral
3.2.4. Alternative Contours: Wedges, Rectangles, and Others – reducing the number of singularities in a contour to simplify calculation
3.2.5. Integrands Having Algebraic Functions and the Residue At Infinity – whole new paradigm in evaluating definite integrals with finite limits of an integrand having branch points at the finite limits, defining the residue at infinity, branch point at infinity
3.3. Sums
3.3.1. Complex Integral Representations – selection of integrand and contour to produce sums, demonstration of convergence of complex integral as contour expands to infinity
3.3.2. Examples – rational summands, summands with trigonometric functions
3.4. Problems
4. Cauchy Principal Value
4.1. Integrands Having Poles On the Contour
4.1.1. Definition of a Cauchy Principal Value – definition as a limit, illustration with simple examples
4.1.2. Managing Divergent Terms of a Contour Integral – detailed illustrations of evaluating definite integrals via complex integrals having contributions with divergent terms that cancel
4.2. Analytic Signals and Hilbert Transforms – equivalence of Cauchy-Riemann equations and Hilbert transforms of real and imaginary parts of an analytic function, illustrations of analytic signals having harmonic real and imaginary parts, examples of deriving imaginary parts of analytic function from real part
4.3. Problems
5. Integral Transforms
5.1. Preliminary Matters
5.1.1. The Dirac Delta Function – derivation via self-transform in Hilbert transform integrals, review of properties
5.1.2. A General Discussion of Integral Transforms - integral transforms require a computable inverse to be of any use, conditions under which inverses exist, general format of integral transforms
5.2. The Fourier Transform
5.2.1. Definition and Plancherel’s Theorem – mean square error, and inner product spaces, the Fourier Transform as a Principal Value
5.2.2. Jordan’s Lemma – evaluating Fourier integrals using complex integration, convergence conditions
5.2.3. Parseval’s Theorem – statement, examples of integral evaluations, Fourier series and application of theorem to sums
5.2.4. Convolution Theorem – statement and derivation, applications
5.2.5. Analyticity of the Fourier Transform In the Complex Plane - theorem relating rates of convergence of Fourier transforms and their inverses in the complex plane, strips of convergence, causality
5.2.6. Poisson Sum Formula – derivation, application to computation of error function to machine precision anywhere in the complex plane
5.3. The Laplace Transform
5.3.1. Definition – extending the discussion of analyticity of the Fourier transform with an exponentially decaying kernel rather than an oscillatory kernel, derivation of inverse as an integral in the complex plane
5.3.2. Convolution Theorem – derivation, examples, application to computing certain classes of definite integrals
5.3.3. Inversion Via Complex Integration
5.3.3.1. Solutions to Ordinary Differential Equations and Rational Transforms – initial conditions, homogeneous and inhomogeneous equations, inversion via the residue theorem
5.3.3.2. Solutions to Partial Differential Equations and Multivalued Transforms – heat equation produces multivalued transforms, evaluation of inverse Laplace transforms to derive solutions
5.4. The Mellin Transform
5.4.1. Definition discussion of strip of convergence, inverse Mellin transform
5.4.2. Convolution Theorem – derivation; NB this will be used in the next chapter
5.4.3. Scaling – expression of scaled integrals in terms of residues
5.5. Problems
6. Asymptotic Analysis
6.1. Definitions
6.1.1. Big-O, Little-O, and The Squiggle – i.e., definitions of asymptotic equivalence in specific limits
6.1.2. Asymptotic Series – definition, properties, numerical calculations, summation acceleration techniques
6.2. Integration by Parts – development of asymptotic series; limitations
6.2.1. Euler-Maclurin Formula – derivation of asymptotic series using integration by parts, application to evaluation of sums
6.3. Watson’s Lemma and h-Transforms – asymptotic behavior of monotonic integrands, application of Mellin transforms in derivation
6.3.1. Application to Complex Integration – evaluation of integrals with branch points at infinity using h-transforms
6.4. Laplace’s Method – asymptotic behavior of nonmonotomic, nonoscillatory integrals
6.5. The Method of Steepest Descents – deriving asymptotic behavior of complex integrals, derive behavior of real integrals by using Cauchy’s Theorem
6.6. Problems
