# Metamath Proof Explorer

## Table of Contents - 14. BASIC REAL AND COMPLEX FUNCTIONS

- Polynomials
- Polynomial degrees
- The division algorithm for univariate polynomials
- Elementary properties of complex polynomials
- The division algorithm for polynomials
- Algebraic numbers
- Liouville's approximation theorem

- Sequences and series
- Taylor polynomials and Taylor's theorem
- Uniform convergence
- Power series

- Basic trigonometry
- The exponential, sine, and cosine functions (cont.)
- Properties of pi = 3.14159...
- Mapping of the exponential function
- The natural logarithm on complex numbers
- Logarithms to an arbitrary base
- Theorems of Pythagoras, isosceles triangles, and intersecting chords
- Solutions of quadratic, cubic, and quartic equations
- Inverse trigonometric functions
- The Birthday Problem
- Areas in R^2
- More miscellaneous converging sequences
- Inequality of arithmetic and geometric means
- Euler-Mascheroni constant
- Zeta function
- Gamma function

- Basic number theory
- Wilson's theorem
- The Fundamental Theorem of Algebra
- The Basel problem (ζ(2) = π
^{2}/6)
- Number-theoretical functions
- Perfect Number Theorem
- Characters of Z/nZ
- Bertrand's postulate
- Quadratic residues and the Legendre symbol
- Gauss' Lemma
- Quadratic reciprocity
- All primes 4n+1 are the sum of two squares
- Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem
- The Prime Number Theorem
- Ostrowski's theorem