Sine in terms of Hyperbolic Sine

Theorem

Let $z \in \C$ be a complex number.

Then:

$i \sin z = \map \sinh {i z}$

where:

$\sin$ denotes the complex sine

$\sinh$ denotes the hyperbolic sine

$i$ is the imaginary unit: $i^2 = -1$.

$\ds \map \sinh {i z}$

$=$

$\ds \frac {e^{i z} - e^{-i z} } 2$

Definition of Hyperbolic Sine

$\ds $

$=$

$\ds i \frac {e^{i z} - e^{-i z} } {2 i}$

multiplying top and bottom by $i$

$\ds $

$=$

$\ds i \sin z$

$\blacksquare$

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $(4.22)$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.80$: Relationship between Hyperbolic and Trigonometric Functions
1969: J.C. Anderson, D.M. Hum, B.G. Neal and J.H. Whitelaw: Data and Formulae for Engineering Students (2nd ed.) ... (previous) ... (next): $4.$ Mathematics: $4.3$ Trigonometric identities and hyperbolic functions
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hyperbolic functions
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hyperbolic functions