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1.21 Hyperbelfunktionen
Definitionen
Grundbeziehungen
Beziehungen zwischen
den Hyperbelfunktionen
(vgl. die entsprechenden
Formeln der trigonometrischen
Funktionen)
Additionstheoreme und
andere Beziehungen
� für x � 0
– für x � 0
− −
ex − e x ex + e x
sinh x = ; coshx=
2 2
tanh x =
Mathematik
Hyperbelfunktionen
x −x 2x x −x
2x
e −e e − 1 e + e e + 1
= ; cothx=
=
x −x 2x x −x
2x
e + e e + 1 e −e e −1
cosh2x− sinh2x = 1 sinh x cosh x
tanh x = ; coth x =
tanh x⋅ coth x = 1 coshx sinh x
sinh x = cosh2 x−
1=
tanh x
1−tanh2x =
1
coth2x−1 cosh x = sinh2 x+
1 =
1
1−tanh2x =
cothx
coth2x−1 sinh x cosh2 x−11
tanh x = = =
sinh2 x + 1 cosh x coth x
sinh2 x+ 1 cosh x 1
coth x = = =
sinh x cosh2 x −1
tanh x
Für negative x gilt:
sinh (– x) = – sinh x tanh (– x) = – tanh x
cosh (– x) = cosh x coth (– x) = – coth x
sinh (x � y) = sinh x · cosh y � cosh x · sinh y
cosh (x � y) = cosh x · cosh y � sinh x · sinh y
tanh x± tanhy 1± coth x⋅cothy tanh (x � y) =
; coth( x± y)
=
1± tanh x⋅ tanhy coth x± cothy
sinh 2 x = 2sinhx⋅ cosh x
2 tanhx
tanh 2 x =
2 1+ tanh x
2 2
cosh 2 x = sinh x+ cosh x
2 1+ coth x
coth 2 x =
2cothx
n
(cosh x± sinh x) = cosh nx± sinh nx
x cosh x−1x cosh x−1sinh x
sinh =± ; tanh = =
2 2 2 sinhx coshx+ 1
x cosh x+ 1 x sinhx cosh x+
1
cosh = ; coth = =
2 2 2 cosh x−1sinh x
sinh x � sinh y = 2 sinh 1
1
(x � y) cosh (x � y)
2 2
cosh x � cosh y = 2 cosh 1
1
(x � y) cosh (x – y)
2 2
cosh x – cosh y = 2 sinh 1
1
(x � y) sinh (x – y)
2 2
sinh( x± y)
tanhx± tanhy
=
cosh xcoshy 25
1