Complex Variables and Transforms

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 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.

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.

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.

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.

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.

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.