Functions, Recursions, and Iterations in Mathematics

Mathematical Areas & Formulas Depending on Functions, Recursion, and Iteration

1. Functions & Functional Equations

Calculus: differentiation and integration of f(x), Taylor series expansions.
Functional Equations: Cauchy’s equation f(x+y)=f(x)+f(y), gamma function defined by \Gamma(x+1)=x\,\Gamma(x).
Generating Functions: ordinary \displaystyle G(a_n; x)=\sum_{n\ge0}a_n x^n and exponential \displaystyle E(a_n; x)=\sum_{n\ge0}a_n\frac{x^n}{n!} to encode sequences and solve recurrences.
Functional Analysis: spaces of functions and operators (e.g., Banach and Hilbert spaces).

Linear Recurrences: Fibonacci F_{n}=F_{n-1}+F_{n-2}, general a_n = c_1\,a_{n-1} + \cdots + c_k\,a_{n-k}.
Factorials & Combinatorics: n! = n\,(n-1)!; Catalan numbers C_{n}=\sum_{i=0}^{n-1}C_i\,C_{n-1-i}.
Difference Equations: discrete analogues of differential equations, e.g. y_{n+1}-y_n = g(n,y_n).
Partition Functions: p(n)=p(n-1)+p(n-2)-p(n-5)-\dots (Euler’s pentagonal number theorem).

Fixed-Point Iteration: x_{k+1}=g(x_k) to solve x=g(x).
Newton–Raphson Method: x_{k+1}=x_k - \frac{f(x_k)}{f'(x_k)} for root-finding.
Gradient Descent: x_{k+1}=x_k - \alpha\nabla f(x_k) to minimize a cost function.
Power Method: v_{k+1}=A\,v_k/\|A\,v_k\| to compute largest eigenvalue of A.

Logistic Map: x_{n+1}=r\,x_n\,(1-x_n) illustrating chaos theory.
Julia & Mandelbrot Sets: iterating z_{n+1}=z_n^2 + c in the complex plane.
Iterated Function Systems: constructing fractals via repeated affine transformations.