Propagators in Quantum Chemistry, 2nd Edition

Appendix A

Complex Calculus Primer

A complex variable z has a real part x and an imaginary part y and is written z = x + iy. Functions f(z) of a complex variable are our concern. Explicit examples are 1/z, z², and exp(z). Any function of a complex variable can be expressed as

where u and v are real functions.

Figure A.l: Complex number plane.

A.1   Continuity

A function f(z) is said to be continuous at z = z₀ if given any positive number , one can find a number δ such that |f(z) − f (z₀)| < provided that |z − z₀| < δ.

Let z₀ = x₀ + iy₀. It then follows that |u (x, y) − u (x₀, y₀)| ≤ |f(z) − f(z₀)| < if |z − z₀| < δ i.e., |x − x₀| < δ, y − y₀| < δ/. Hence, if f(z) is continuous, so is u(x, y) and similarly v(x, y). Conversely, it can be shown that if u(x, y) and v(x, y) are continuous, so ...