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Axioma

Mathematics · Frontier · 30 min · Chapter 16 of 28

Complex analysis

Calculus over the complex numbers. Differentiability once, in the complex sense, is so strong that it creates infinite smoothness and the residue calculus.

Read first: Real analysis, Trigonometry

The idea

Write z = x + iy with x, y real and i² = −1. Addition is vector addition. Multiplication has the rule i² = −1, and geometrically it multiplies lengths and adds arguments. The conjugate x − iy reflects across the real axis, and |z|² = z times its conjugate. Every nonzero complex number has an inverse, namely the conjugate divided by the squared modulus. The complex numbers are a field that cannot be ordered in a way compatible with the arithmetic, because squares can be negative.

Euler’s formula e^{iθ} = cos θ + i sin θ is the statement that the exponential map sends the imaginary axis onto the unit circle. It packages the angle-sum formulas: e^{i(α+β)} = e^{iα} e^{iβ}. De Moivre’s formula (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) is the same fact.

A function is holomorphic at a point when the complex derivative exists in a neighborhood. That single requirement, a limit of [f(z + h) − f(z)] / h as the complex increment h → 0, is far stronger than two real partial derivatives. Holomorphic functions are analytic: they equal their Taylor series. Cauchy's integral formula recovers the value inside a contour from the values on the contour, f(a) = (1 / 2πi) ∮ f(z)/(z − a) dz. If a function has isolated singularities, the integral around a large contour equals 2πi times the sum of residues, the coefficients of 1/(z − a) in the local Laurent series. That is a practical machine for definite integrals.

Hold onto this

  1. A complex number x + iy is a point in the plane, and multiplying by i rotates a quarter turn.
  2. Euler’s formula identifies e^{iθ} with the point (cos θ, sin θ) on the unit circle.
  3. Complex differentiability on a region implies a power series expansion there.
  4. The residue theorem evaluates real integrals by summing coefficients of 1/(z − a).

Formulas

Euler’s formula

The exponential of a pure imaginary number lies on the unit circle.

θ
A real angle, in radians

Modulus

Distance from the origin in the complex plane.

x, y
Real and imaginary parts

Cauchy’s integral formula

A holomorphic function inside and on a simple closed curve is determined by its boundary values.

γ
A simple closed curve, positively oriented, with a inside

Residue theorem

The integral is 2πi times the sum of residues at singularities inside the contour.

Res(f, a)
Coefficient of 1/(z − a) in the Laurent series at a

Worked examples

1. A power on the unit circle

Compute (1 + i)² in two ways.

  1. Algebra: 1 + 2i + i² = 1 + 2i − 1 = 2i.
  2. Polar: 1 + i has modulus √2 and argument π/4. Squaring doubles the argument and squares the modulus: modulus 2, argument π/2, which is the point 2i.

2i. The two calculations agree.

2. Euler at a right angle

What is e^{iπ/2}?

  1. cos(π/2) + i sin(π/2) = 0 + i·1.

i. And e^{iπ} = −1, which is Euler’s identity e^{iπ} + 1 = 0.

Try these

Find the residue of 1/z at 0, and the integral of 1/z around the unit circle.

Express cos θ using complex exponentials.

Common mistakes

  • Complex differentiability is not the same as differentiability of the two real component functions. The Cauchy–Riemann equations must hold.
  • A branch of the logarithm, or of a square root, cannot be holomorphic on the whole punctured plane. You have to cut the plane.
  • The orientation of the contour matters. Clockwise reverses the sign of the integral.