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Derivative identities for the real restrictions of complex sine and cosine

Statement

For $t \in R$ , define $S (t) = Re ((E (i t) - E (- i t)) / (2 i))$ and $C (t) = Re ((E (i t) + E (- i t)) / 2)$ , where $E (z) = \sum_{n = 0}^{\infty} z^{n} / n!$ and $Re (x + i y) = x$ . Then $S$ and $C$ are differentiable on $R$ and satisfy $S^{'} (t) = C (t)$ and $C^{'} (t) = - S (t)$ for every real $t$ .

Statement references

The complex sine and cosine derivative formulas and the real restriction: Real sine and cosine as real restrictions of complex trigonometric functions.

Complex sine and cosine are entire with derivatives $s_{C}^{'} = c_{C}$ and $c_{C}^{'} = - s_{C}$ : Complex sine, cosine, hyperbolic sine, and hyperbolic cosine are entire with their standard derivatives.

Complex differentiability implies real total differentiability with the corresponding real component derivatives: Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations.

The power-series definitions and termwise differentiation: Stein and Shakarchi, Complex Analysis, Chapter 1, §2.3, printed pp. 16–17 (PDF pp. 34–35).

Facts & Assumptions

Given: A real number $t$ .

[F1]

The functions $S (t)$ and $C (t)$ are the real parts of the complex sine and cosine defined by the exponential formulas in Real sine and cosine as real restrictions of complex trigonometric functions.

[F2]

The complex sine and cosine are entire and have derivatives $s_{C}^{'} (z) = c_{C} (z)$ and $c_{C}^{'} (z) = - s_{C} (z)$ (Complex sine, cosine, hyperbolic sine, and hyperbolic cosine are entire with their standard derivatives).

[F3]

Let $U \subseteq C$ be open, let $z = x + i y \in U$ , and let $f = u + i v : U \to C$ . If $f$ is complex differentiable at $z$ , then its real total derivative exists, $u_{x} (x, y) = Re (f^{'} (z))$ , and $v_{x} (x, y) = Im (f^{'} (z))$ (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations). Both applications below have $U = C$ , which is open and contains every real $t$ , because the functions in [F2] are entire.

Proof

1.1givenF1F2F3

Let $s_{C} = u + i v$ . By [F2] it is complex differentiable at every real $t$ with derivative $c_{C} (t)$ . By [F3], $u$ is real totally differentiable at $(t, 0)$ and $u_{x} (t, 0) = Re (c_{C} (t)) = C (t)$ . Restricting the real total-differentiability expansion to increments $(h, 0)$ gives $S^{'} (t) = u_{x} (t, 0) = C (t)$ .

2.1givenF1F2F3∎

Let $c_{C} = w + i z$ . By [F2] its complex derivative at a real $t$ is $- s_{C} (t)$ . By [F3], $w$ is real totally differentiable at $(t, 0)$ and $w_{x} (t, 0) = Re (- s_{C} (t)) = - S (t)$ . Restricting to increments $(h, 0)$ gives $C^{'} (t) = w_{x} (t, 0) = - S (t)$ .

Depends on

Dependency tree · 2 levels

Sources

Elias M. Stein and Rami Shakarchi, Complex Analysis, Chapter 1, §§2.2–2.3