Periodicity of Hyperbolic Sine

Theorem

Let $k \in \Z$.

Then:

$\map \sinh {x + 2 k \pi i} = \sinh x$

Proof

\(\ds \map \sinh {x + 2 k \pi i}\)

\(=\)

\(\ds \frac {e^{x + 2 k \pi i} - e^{-\paren {x + 2 k \pi i} } } {2 i}\)

Definition of Hyperbolic Sine

\(\ds \)

\(=\)

\(\ds \frac {e^{i \paren {-i x + 2 k \pi} } - e^{i \paren {i x + 2 \paren {-k} \pi} } } {2 i}\)

$i^2 = -1$ and simplifying

\(\ds \)

\(=\)

\(\ds \frac {e^{i \paren {-i x} } - e^{i \paren {i x} } } {2 i}\)

Periodicity of Complex Exponential Function

\(\ds \)

\(=\)

\(\ds \frac {e^x - e^{- x} } {2 i}\)

$i^2 = -1$

\(\ds \)

\(=\)

\(\ds \sinh x\)

Definition of Hyperbolic Sine

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.86$: Periodicity of Hyperbolic Functions