maths.free › Complex Analysis › Holomorphic functions and the Cauchy-Riemann equations

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).

Ejemplo resuelto · derivative of x^2 - y^2 wrt x

Calculus · Derivatives

Differentiate x^2 - y^2

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

Paso a paso

  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.

Revelar la respuesta
f'(x) = 2 x

Ahora tú Escoge un problema, o escribe o dibuja el tuyo. Cada paso, una imagen, la respuesta oculta hasta que preguntes.

Guarda tu propio trabajo

Una cuenta gratuita añade notas sobre cada lección, un registro de lo que has terminado, tus problemas resueltos en un lugar, y un tutor que puedes preguntar sobre esta página. Las matemáticas en sí están abiertas a todos, se han registrado o no.

Registrarse Inicio de sesión

Símbolos utilizados aquí

Toque cualquier símbolo para la definición completa, una imagen, y lo que cada letra en ella significa.

Cómo: 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.

Preguntas que la gente hace

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.

Más en Complex Analysis

The complex plane and Euler's formulaContour integrals and the residue theoremPower series and analytic continuation