CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Complex sine, cosine, hyperbolic sine, and hyperbolic cosine are entire with their standard derivatives

Statement

The functions $\sin, \cos, \sinh, \cosh$ are entire and satisfy $\sin^{'} = \cos, \cos^{'} = - \sin, \sinh^{'} = \cosh, \cosh^{'} = \sinh .$

Facts & Assumptions

Given: The four entire power series of The exponential definitions of complex sine, cosine, hyperbolic sine, and hyperbolic cosine equal their entire power series.

[L1]

A complex power series may be differentiated term by term inside its radius (Inside its disc of convergence a complex power series is holomorphic and may be differentiated term by term).

Proof

technique · direct

1.1L1algebra

Apply [L1] to each of the four infinite-radius series and cancel the positive integer factor against the factorial.

2.1step 1.1∎

Shifting the resulting indices gives respectively the series for $\cos, - \sin, \cosh, \sinh$ . Infinite radius makes each function entire.

Depends on

Used by

The Basel sum is pi squared over six by a residue computation Corollary
The partial-fraction expansion of pi-squared cosecant-squared Corollary
Agreement accumulating only at the boundary does not force a holomorphic identity Counterexample
sin z-z has a zero of order three at the origin Example
The circle integral of cos z/(z-1)³ over |z|=2 is -π i cos 1 Example
FALSE: an entire function bounded on the real axis is constant False statement
Cosecant residues sum an alternating rational series over the integers Theorem
Cotangent residues sum a rational function over the integers Theorem
The complex Pythagorean identity by the identity theorem Theorem
The Mittag-Leffler expansion of pi cotangent Theorem

Dependency tree · two levels

10 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

E. Stein and R. Shakarchi, Complex Analysis, Ch. 1 (standard reference, not scraped)