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$z \mapsto z^{2}$ is entire with derivative $2 z$ , directly from the complex difference quotient

Example

The square function $f (z) = z^{2}$ is entire and satisfies $f^{'} (z) = 2 z$ . In Cartesian coordinates its components are $u (x, y) = x^{2} - y^{2}, v (x, y) = 2 x y,$ and both are harmonic; $v$ is a harmonic conjugate of $u$ .

Facts & Assumptions

Given: An arbitrary $z \in C$ .

[L1]

A function is complex differentiable at $z$ when its punctured-domain difference quotient has a complex limit there, and it is entire when it is complex differentiable at every point of $C$ (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions).

[L2]

The complex modulus is multiplicative, satisfies the triangle inequality, and is definite (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[L3]

For a holomorphic function with $C^{2}$ components, both components satisfy Laplace's equation, and the imaginary component is a harmonic conjugate of the real component (The $C^{2}$ real and imaginary parts of a holomorphic function satisfy Laplace's equation and form a harmonic-conjugate pair).

Verification

technique · direct computation

1.1

For every nonzero increment $h$ , $\frac{f (z + h) - f (z)}{h} = \frac{(z + h)^{2} - z^{2}}{h} = 2 z + h .$

algebra

2.1

Since $∣ (2 z + h) - 2 z ∣ = ∣ h ∣ \to 0$ , the quotient in step 1.1 tends to $2 z$ . Thus $f^{'} (z) = 2 z$ by [L1].

step 1.1L1L2

3.1

The point $z$ was arbitrary, so $f$ is entire.

step 2.1L1

3.2

Writing $z = x + i y$ gives $z^{2} = (x^{2} - y^{2}) + i (2 x y)$ . Hence $u_{x} = 2 x = v_{y}$ and $u_{y} = - 2 y = - v_{x}$ , in agreement with step 2.1.

step 2.1algebra

4.1

Moreover $u_{x x} + u_{y y} = 2 - 2 = 0$ and $v_{x x} + v_{y y} = 0 + 0 = 0$ . The polynomial components are $C^{2}$ , so [L3] identifies them as harmonic and identifies $v$ as a harmonic conjugate of $u$ .

step 3.1step 3.2L3algebra∎

Depends on

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Sources

MIT 18.04, Topic 2: Functions of a Complex Variable (standard reference, not scraped)

z↦z2z↦z2 is entire with derivative 2z2z, directly from the complex difference quotient