Complex numbers extend the real number line to a two-dimensional plane by introducing the imaginary unit i, defined by i^2 = -1. A complex number z is expressed as z = a + bi, where a and b are real numbers.
Formally, the field of complex numbers, \mathbb C, can be constructed as the quotient ring R[x]/(x^2 + 1), which makes it clear that \mathbb C is the algebraic closure of the real numbers.
This is the statement of the Fundamental Theorem of Algebra: every non-constant single-variable polynomial with complex coefficients has at least one complex root.
Geometrically, complex numbers are represented on the Argand plane, allowing for a polar representation z = r \, e^{i\theta} via Euler's formula.
This representation is used for describing rotations and oscillations, and it forms the basis of complex analysis, the study of functions on the complex plane (holomorphic functions).
The Riemann zeta function is a function of a complex variable, initially defined as an infinite series. It has connections to number theory, particularly in understanding the distribution of prime numbers. This section here explains the function and its properties, including the Riemann Hypothesis.
SVÉSNIKOV, Aleksey G. and TICHONOV, Andrej N., 1984. Teoria delle funzioni di una variabile complessa. Roma: Editori Riuniti Univ. Press. ISBN 978-88-359-2800-3.
BRUNTON, Steve, ME 564 - Mechanical Engineering Analysis, University of Washington.