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$z \mapsto 1 / z$ is holomorphic on $C ∖ {0}$ with derivative $- 1 / z^{2}$ , directly from the difference quotient

Example

The reciprocal function $f (z) = 1 / z$ is holomorphic on the punctured plane $C ∖ {0}$ , and $f^{'} (z) = - \frac{1}{z^{2}} .$

Facts & Assumptions

Given: A point $z \in C ∖ {0}$ .

[L1]

Complex differentiability is the existence of the punctured-domain difference-quotient limit, and holomorphy on an open set means complex differentiability at each point of that set (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions).

[L2]

The complex modulus is multiplicative, satisfies $∣ z + w ∣ \leq ∣ z ∣ + ∣ w ∣$ , and vanishes exactly at $0$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive). Applying the triangle inequality to $z = (z + h) + (- h)$ , and using $∣ - h ∣ = ∣ - 1 ∣ ∣ h ∣ = ∣ h ∣$ , gives the reverse form $∣ z + h ∣ \geq ∣ z ∣ - ∣ h ∣$ .

Verification

technique · direct computation

1.1

If $0 < ∣ h ∣ < ∣ z ∣ / 2$ , then $∣ z + h ∣ \geq ∣ z ∣ - ∣ h ∣ > ∣ z ∣ / 2 > 0$ , so $z + h$ remains in the punctured plane.

2.1

For such $h$ , $\frac{f (z + h) - f (z)}{h} = - \frac{1}{z (z + h)} .$

step 1.1algebra

3.1

The error from the proposed derivative satisfies $∣ - \frac{1}{z (z + h)} + \frac{1}{z^{2}} ∣ = \frac{∣ h ∣}{∣ z ∣^{2} ∣ z + h ∣} \leq \frac{2 ∣ h ∣}{∣ z ∣^{3}} ⟶ 0.$

step 1.1step 2.1L2algebra

4.1

Hence $f^{'} (z) = - 1 / z^{2}$ by [L1]. Since $z$ was arbitrary in $C ∖ {0}$ , $f$ is holomorphic there. The point $0$ is absent from the function's domain, not merely exceptional for the derivative formula.

step 3.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 36 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Guide to Cultivating Complex Analysis (standard reference, not scraped)

z↦1/zz↦1/z is holomorphic on C∖{0}C∖{0} with derivative −1/z2−1/z2, directly from the difference quotient