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hyperbolic

# hyperbolic - ix = 1 2 e ix e − ix = 1 2 cos x i sin x cos...

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Hyperbolic Functions Definitions sinh x = 1 2 e x e x ( ) = hyperbolic sine function cosh x = 1 2 e x + e x ( ) = hyperbolic cosine function tanh x = sinh x cosh x = e x e x e x + e x = hyperbolic tangent function Identities cosh 2 x sinh 2 x = 1 sinh 2 x = 1 2 1 cosh 2 x ( ) [ ] cosh 2 x = 1 2 1 + cosh 2 x ( ) [ ] sinh 2 x ( ) = 2sinh x cosh x cosh 2 x ( ) = 2cosh 2 x 1 1 coshx sinhx tanhx -1

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Relation of Hyperbolic Functions to Harmonic Trig Functions sinh ix ( ) = 1 2 e ix e ix ( ) = 1 2 cos x + i sin x cos x i sin x ( ) [ ] = i sin x sin x = i sinh ix ( ) OR sinh x ( ) = i sin ix ( ) cosh
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Unformatted text preview: ix ( ) = 1 2 e ix + e − ix ( ) = 1 2 cos x + i sin x + cos x − i sin x ( ) [ ] = cos x ⇒ cos x = cosh ix ( ) OR cosh x = cos ix ( ) tanh ix ( ) = e ix − e − ix e ix + e − ix = cos x + i sin x − cos x − i sin x ( ) cos x + i sin x + cos x − i sin x ( ) = i sin x cos x ⇒ tan x = − i tanh ix ( ) OR tanh x ( ) = − i tan ix ( )...
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hyperbolic - ix = 1 2 e ix e − ix = 1 2 cos x i sin x cos...

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