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Derivative of a complex oscillatory phase

Statement

If $a < b$ , $g \in C^{1} [a, b]$ is real-valued and $ω \in R$ , then $x \mapsto e^{i ω g (x)}$ is continuously differentiable and $\frac{d}{d x} e^{i ω g (x)} = i ω g^{'} (x) e^{i ω g (x)} .$

Facts & Assumptions

Given: $a < b$ , a real $C^{1}$ function $g$ on $[a, b]$ , and a real number $ω$ .

[F1]

$\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ : for real $y$ , $e^{i y} = \cos y + i \sin y$ .

[F2]

The derivatives of sine and cosine are cosine and minus sine: $(\sin y)^{'} = \cos y$ and $(\cos y)^{'} = - \sin y$ .

[F3]

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ : if $v : A \to B \subseteq R$ is differentiable at a limit point $x \in A$ of $A$ , and $u : B \to R$ is differentiable at the limit point $v (x)$ of $B$ , then $(u \circ v)^{'} (x) = u^{'} (v (x)) v^{'} (x)$ .

Proof

1.1givenF1construct

Set $θ (x) = ω g (x)$ . By [F1], $e^{i θ (x)} = \cos (θ (x)) + i \sin (θ (x))$ .

2.1givenF2F3step 1.1algebra∎

The scalar difference quotient gives $θ^{'} (x) = ω g^{'} (x)$ . Apply [F3] with $A = [a, b]$ , $B = R$ , $v = θ$ , and $u = \sin$ or $\cos$ : every $x \in [a, b]$ is a limit point of $[a, b]$ since $a < b$ , and $θ (x)$ is a limit point of $R$ , where both outer functions are differentiable by [F2]. Thus [F2] and step 1.1 give the componentwise derivative $- ω g^{'} (x) \sin (θ (x)) + i ω g^{'} (x) \cos (θ (x)) = i ω g^{'} (x) e^{i θ (x)}$ , including the one-sided endpoint derivatives. Continuity follows from continuity of $g^{'}$ , $g$ , and the trigonometric functions.

Depends on

Used by

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

NIST Digital Library of Mathematical Functions, §2.3 (standard reference, not scraped)
A. Iserles, S. P. Nørsett, and S. Olver, Highly oscillatory quadrature: The story so far, §3.3 (standard reference, not scraped)