This is meant to be a short collection of mathematical functions implemented in GO. following the list of the functions already implemented:
- DFT
- Laplace
- FFT
DFT converts a sequence of N real or complex numbers into N complex frequency components, revealing a signal’s frequency content. It uses the formula ( X_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i k n / N} ), where ( x_n ) is the input signal. Computationally intensive (O(N²)), it’s key for signal processing tasks like audio or image analysis. In Go, it’s implemented using complex128 for calculations, with applications in spectrum analysis and filtering.
The Laplace Transform converts a time-domain function ( f(t) ) into the s-domain via ( F(s) = \int_0^\infty f(t) e^{-st} dt ), where ( s ) is complex. It’s used in control systems and differential equations to analyze system stability. In Go, numerical approximation (e.g., Riemann sum) computes it for arbitrary functions, using complex128. It’s vital for engineering and physics simulations.
FFT is an efficient algorithm for computing the DFT, reducing complexity from O(N²) to O(N log N). It splits the input (length must be a power of 2) into even/odd indices recursively (Cooley-Tukey method). In Go, it uses complex128 for frequency components, ideal for real-time signal processing like audio or image analysis.