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Holomorphic functions and the Cauchy–Riemann equations

What complex differentiability demands, and why it is so restrictive.

f = u + iv is holomorphic when u_x = v_y and u_y = −v_x. Then u and v are harmonic and f is infinitely differentiable. Picture it: z² = (x² − y²) + i(2xy): the level curves of the real and imaginary parts cross at right angles. Think it: the derivative at a point is a rotation-and-scaling, so holomorphic maps preserve angles (conformal).

Вработен пример: derivative of x^2 - y^2 wrt x

Calculus · Derivatives

Differentiate x^2 - y^2

\frac{d}{dx}\left[x^{2} - y^{2}\right]

Чекор по чекор

  1. \frac{d}{dx}\left[x^{2} - y^{2}\right]

    Start from the derivative to compute.

  2. \frac{d}{d x} x^{2} + \frac{d}{d x} \left(- y^{2}\right)

    Sum rule: differentiate term by term.

  3. 2 x + \frac{d}{d x} \left(- y^{2}\right)

    Power rule: (xⁿ)′ = n·xⁿ⁻¹ with n = 2.

  4. 2 x

    The derivative of a constant is 0.

Одкриј го одговорот
f'(x) = 2 x

Symbols used here

f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
i
imaginary unit
i² = −1.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
\theta
theta
The usual name for an angle.
\frac{\partial f}{\partial x}
partial derivative
Derivative with respect to x, holding the other variables fixed.
\iint,\ \oint
double / contour integral
Integral over a region of the plane; integral around a closed curve.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
\zeta(s)
Riemann zeta function
Σ 1/nˢ, continued to the complex plane; its zeros are the Riemann Hypothesis.
\operatorname{Res}_{z=a} f
residue
Coefficient of 1/(z − a) at a singularity; drives contour integrals.
f: \mathbb{C} \to \mathbb{C},\ \bar{z}
complex function, conjugate
A map of the complex plane; x − iy.

How to: Holomorphic functions and the Cauchy–Riemann equations

  1. Start from the derivative to compute.
  2. Sum rule: differentiate term by term.
  3. Power rule: (xⁿ)′ = n·xⁿ⁻¹ with n = 2.
  4. The derivative of a constant is 0.

Questions people ask

Why is complex differentiability so much stronger than real?

The limit must be the same from every direction in the plane, not just two. That forces the Cauchy–Riemann equations, which in turn force infinitely many derivatives and a convergent Taylor series.

What is a residue?

The coefficient of 1/(z − a) in the Laurent series at a singularity a. The residue theorem says a contour integral equals 2πi times the sum of the residues inside, which evaluates many real integrals in one line.

Обиди се со себе.

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