Hyperbolic Sine in terms of Sine

Theorem

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

Then:

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

where:

$\sin$ denotes the complex sine

$\sinh$ denotes the hyperbolic sine

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

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

$=$

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

$\ds $

$=$

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

$i^2 = -1$

$\ds $

$=$

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

$i^2 = -1$

$\ds $

$=$

$\ds i \sinh z$

Definition of Hyperbolic Sine

$\blacksquare$

This identity is also seen in the form:

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

which can be seen to follow from the other form by multiplication by $-i$.

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: $8.74$: 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: $5$
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): hyperbolic function