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).

Vinna dæmi · derivative of x^2 - y^2 wrt x

Calculus · Derivatives

Differentiate x^2 - y^2

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

Skref fyrir skref

\frac{d}{dx}\left[x^{2} - y^{2}\right]
Start from the derivative to compute.
\frac{d}{d x} x^{2} + \frac{d}{d x} \left(- y^{2}\right)
Sum rule: differentiate term by term.
2 x + \frac{d}{d x} \left(- y^{2}\right)
Power rule: (xⁿ)′ = n·xⁿ⁻¹ with n = 2.
2 x
The derivative of a constant is 0.

Sýna svarið

f'(x) = 2 x

Núna ūú Veldu vandamál, eða sláðu inn eða teikna eigin. Hvert skref, mynd, svarið falið þar til þú spyrð.

Haltu eigin vinnu þinni

Ókeypis reikningur bætir við athugasemdum við hverja kennslustund, skrá yfir það sem þú hefur lokið, leyst vandamál á einum stað og kennari sem þú getur spurt um þessa síðu.Stærðfræðin sjálf er opin öllum, skráður inn eða ekki.

Skráðu þig inn Innskráning

Tákn notuð hér

Pikkaðu á hvaða tákn sem er fyrir fulla skilgreiningu, mynd og hvað hvert bréf í því merkir.

Hvernig á að nota: Holomorphic functions and the Cauchy-Riemann equations

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

Spurningar sem fólk spyr

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.

Meira í Complex Analysis

The complex plane and Euler's formula Contour integrals and the residue theorem Power series and analytic continuation