Complex Variables and Transforms

Explore complex conjugate properties, including double conjugates, conjugation of sums and products, and extracting real and imaginary parts using z+z̄ and (z - z̄)/(2i) with modulus considerations and worked exercises.

Explore de Moivre's formula and exponentials to convert complex numbers, compute arguments, and solve product and quotient problems with practice exercises.

Discover how to compute the cube roots and general nth roots of complex numbers by converting to exponential form, using principal arguments and the infinite set of angles.

Investigate functions of complex variables by mapping z to w = f(z), analyzing domain and range, and expressing u(x,y) and v(x,y) in rectangular and polar forms.

Explore how trigonometric functions extend to complex numbers by defining sine and cosine via exponential forms with complex arguments, and apply derivatives, identities, and periodicity to complex variables.

Explore hyperbolic functions and their numerical problems, derive identities and derivatives, and learn key relationships between hyperbolic and trigonometric functions for problem solving.

Explore hyperbolic functions of a complex variable, derive key identities, and differentiate hyperbolic functions. Solve examples and exercises to connect hyperbolic and trigonometric functions in complex analysis.

Explore logarithm of complex variables as a multi-valued function, defined by log z = ln r + i(θ + 2πn), and distinguish principal log as its version.

Learn complex logarithms, principal log and arguments, prove log identities for products, and derive inverse trigonometric and hyperbolic functions from logarithms with worked examples.

Review key inverse trigonometric and hyperbolic formulas for complex variables, work through a sine inverse example, compare inverse sine, cosine, and tangent, and note derivative rules for complex forms.

solve exercise problems on inverse trigonometric and hyperbolic functions for complex numbers. apply log-based formulas and the principal argument to evaluate these inverses.

Examine differentiating complex valued functions w(t)=u(t)+iv(t) via real and imaginary parts, establish w'(t)=u'(t)+iv'(t), and introduce definite integration of complex functions with real and imaginary components.

Split the complex-valued function W(t) into its real and imaginary parts and apply even symmetry about the y-axis to show ∫_{-a}^{a} W(t) dt = 2 ∫_{0}^{a} W(t) dt.

Explore analyticity of complex functions using the Cauchy-Riemann equations and apply two integration methods on contours, including the unit circle, for analytic and non-analytic cases.

Apply the first or second method for complex integration by testing analyticity. Solve problems via straight-line shortest paths, parameterize z(t), and compute real and imaginary parts.

Explore Cauchy's integral theorem for analytic functions in simply connected domains; show contour integrals over simple closed paths vanish and the result is independent of the path.

Test analyticity to apply Cauchy's integral theorem and formula, distinguish analytic from non-analytic functions, and use contour circles to evaluate closed-path integrals without direct integration.

Explore sequences and series of complex numbers, define convergence via limits, discuss partial sums, and distinguish convergent and divergent cases with real and imaginary parts.

Derive and verify limits of sequences using definitions, apply epsilon arguments, and establish conjugate and sum properties for complex series via partial sums and remainder.

Explore Taylor and Maclaurin series for complex numbers, using analyticity inside a disc to express functions by derivatives over factorials, with examples from the exponential and sine functions.

Work through Taylor and Maclaurin series for hyperbolic functions, using trig-hyperbolic identities and complex substitution to derive the expansions, and discuss convergence and applicability in the complex plane.

Explore more worked examples of Taylor and Maclaurin series, including hyperbolic functions and the exponential, verify expansions, and apply geometric series techniques to problems with index shifting.

Define harmonic functions as real-valued functions of two variables that satisfy Laplace's equation. Connect harmonic components to analytic functions and note physical examples like temperature and electrostatic potential.

Learn how to verify a real-valued function is harmonic, find its harmonic conjugate, and build the analytic function from its real and imaginary parts.

Discover how any periodic signal can be expressed as a Fourier series of sines and cosines, with coefficients computed over a complete period and extended to general periods.

Explore continued Fourier series for a triangular wave, deriving piecewise definitions and cosine coefficients, showing even symmetry reduces sine terms and assembling the final series.

Explains how even and odd function symmetry about the y-axis and origin informs half-range Fourier series with sine or cosine terms.

Explore the exponential form of the Fourier series to express periodic signals as sums of complex exponentials, with positive and negative frequency terms.

Explore the Fourier transform, linking time-domain signals to their frequency components, and distinguish periodic from aperiodic signals with examples and inverse transforms.

Explore the properties of the Fourier transform, including even/odd decomposition, linearity, time scaling and shifting, frequency shifting, duality, and the convolution relationships between time and frequency domains.

Explore how to recover a time signal from its Fourier transform using the inverse Fourier transform, and apply Parseval's theorem to equate time-domain energy with frequency-domain energy.

Learn bilateral and unilateral Laplace transforms, derive transforms from time signals, and determine the region of convergence in the complex frequency plane.

Discuss inverse Laplace transform through example problems, using table and partial fractions; examine region of convergence, poles and zeros, unilateral and bilateral transforms, and final and initial value theorems.

Explore unilateral Laplace transform properties—shifting, time-scaling, differentiation, and integration—and solve differential equations with initial conditions using system analysis and complex frequency concepts.

Learn how continuous signals become discrete through sampling, using impulse sampling and delta functions, then apply the z-transform to analyze and recover discrete-time signals via X(z) and its inverse.