Hyperbolic Sine of Complex Number
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Theorem
Let $a$ and $b$ be real numbers.
Let $i$ be the imaginary unit.
Then:
- $\sinh \paren {a + b i} = \sinh a \cos b + i \cosh a \sin b$
where:
- $\sin$ denotes the real sine function
- $\cos$ denotes the real cosine function
- $\sinh$ denotes the hyperbolic sine function
- $\cosh$ denotes the hyperbolic cosine function.
Proof 1
| \(\ds \sinh \paren {a + b i}\) | \(=\) | \(\ds \sinh a \cosh \paren {b i} + \cosh a \sinh \paren {b i}\) | Hyperbolic Sine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sinh a \cos b + \cosh a \sin \paren {b i}\) | Cosine in terms of Hyperbolic Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sinh a \cos b + i \cosh a \sin b\) | Sine in terms of Hyperbolic Sine |
$\blacksquare$
Proof 2
| \(\ds \sinh a \cos b + i \cosh a \sin b\) | \(=\) | \(\ds \frac {e^a - e^{-a} } 2 \frac {e^{i b} + e^{-i b} } 2 + i \frac {e^a + e^{-a} } 2 \frac {e^{i b} - e^{-i b} } {2 i}\) | Definition of Hyperbolic Sine, Euler's Cosine Identity, Definition of Hyperbolic Cosine, Euler's Sine Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{a + i b} - e^{-a + i b} + e^{a - i b} - e^{-a - i b} + e^{a + i b} + e^{-a + i b} - e^{a - i b} - e^{-a - i b} } 4\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{\paren {a + b i} } - e^{-\paren {a + b i} } } 2\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sinh \paren {a + b i}\) | Definition of Hyperbolic Sine |
$\blacksquare$