Module functions 

Auto-generated module

🤖 Generated with SplitRS

Functions

cauchy_riemann_check

Check whether a function f: ℂ → ℂ satisfies the Cauchy-Riemann equations at point z using finite differences with step h.

cayley_transform

Cayley transform: maps the upper half-plane to the unit disk.

cayley_transform_inverse

Inverse Cayley transform: maps the unit disk to the upper half-plane.

complex_exp_taylor

Taylor series approximation of exp(z) = Σ z^k / k! up to terms terms.

complex_gaussian_elimination

Solve the complex linear system A·x = b using Gaussian elimination with partial pivoting.

complex_sin_taylor

Taylor series approximation of sin(z) = Σ (-1)^k z^(2k+1) / (2k+1)!

complex_step_derivative

Numerically differentiate f at z using the complex-step method.

contour_integral_circle

Numerically integrate f(z) along a circular contour of radius r centered at center, using n_points quadrature points.

dct_i

Compute the DCT-I of a real signal.

dct_ii

Compute the DCT-II of a real signal (the “standard” DCT used in JPEG/MP3).

dft_matrix

Compute the N×N DFT matrix W where W[j][k] = exp(-2πi·j·k/N).

durand_kerner_roots

Find all roots of a monic polynomial using the Durand-Kerner (Weierstrass) method.

exponential_conformal

Exponential conformal map: w = exp(z).

fft_convolve

Convolve two real signals using FFT.

fft_radix2

Cooley-Tukey Radix-2 FFT (in-place, decimation-in-time).

hamming_window

Generate a Hamming window of length n.

hann_window

Generate a Hann window of length n.

idct_ii

Compute the inverse DCT-II (DCT-III).

ifft_radix2

Inverse FFT (Cooley-Tukey Radix-2).

istft

Compute the inverse STFT (overlap-add reconstruction).

joukowski

Joukowski conformal map: w = z + 1/z.

laplace_exp_decay

Evaluate the Laplace transform of an exponentially decaying signal f(t) = x * exp(-alpha * t) at complex frequency s.

logarithmic_conformal

Logarithmic conformal map: w = ln(z) (principal branch).

magnitude_spectrum

Compute the magnitude spectrum of a real signal.

naive_dft

Apply the DFT matrix to a vector (O(N²) naive DFT).

parseval_check

Compute Parseval’s theorem check: sum of |x[n]|² == (1/N) sum |X[k]|².

phase_spectrum

Compute the phase spectrum of a real signal.

poly_derivative

Compute the derivative of a polynomial with complex coefficients.

poly_eval

Evaluate a polynomial with complex coefficients at complex point z.

poly_multiply

Multiply two polynomials with complex coefficients.

power_spectral_density

Compute the power spectral density (PSD) of a real signal.

primitive_root_of_unity

Returns the primitive N-th root of unity: exp(2πi/N).

rectangular_window

Generate a rectangular (boxcar) window of length n.

residue_by_contour

Numerically compute the residue of f at pole using the contour integral.

rfft

Compute the DFT of a real-valued signal (returns N/2+1 complex frequencies).

roots_of_unity

Returns the N-th roots of unity: exp(2πi·k/N) for k=0,…,N-1.

schur_2x2

Schur-like decomposition of a 2×2 complex matrix.

schwarz_christoffel_rectangle

Schwarz-Christoffel map approximation for a rectangle.

stft

Compute the Short-Time Fourier Transform of a real signal.

z_transform_eval

Evaluate the Z-transform of a finite-length sequence at complex frequency z.