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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).

Àwọn ààtò àwọn ìṣàmúlò-ètò · derivative of x^2 - y^2 wrt x

Calculus · Derivatives

Differentiate x^2 - y^2

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

Àwọn ìṣàmúlò-ètò

\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.

Fi àwọn àgbèwọlé hàn

f'(x) = 2 x

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Bii o ṣe le: 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.

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Complex Analysis

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