„Breiðbogafall“: Munur á milli breytinga

[[Mynd:csch sech coth.svg|256px|thumb|<font color=#b30000>csch</font>, <font color=#00b300>sech</font> og <font color=#0000b3>coth</font>]]

Skilgreiningar breiðbogafalla eru eftirfarandi, þar sem ''i'' er [[þvertala]]n:

:<math>\sinh x = \frac{e^x - e^{-x}}{2} = -i \sin ix. \!</math>

 

::<math>\cosh x = \frac{e^{x} + e^{-x}}{2} = \cos ix. \!</math>

 

::<math>\tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = \frac{e^{2x} - 1} {e^{2x} + 1} = -i \tan ix. \!</math>

 

::<math>\coth x = \frac{\cosh x}{\sinh x} = \frac {e^x + e^{-x}} {e^x - e^{-x}} = \frac{e^{2x} + 1} {e^{2x} - 1} = i \cot ix. \!</math>

 

::<math>\operatorname{sech} x = \frac{1}{\cosh x} = \frac {2} {e^x + e^{-x}} = \sec ix. \!</math>

 

::<math>\operatorname{csch} x = \frac{1}{\sinh x} = \frac {2} {e^x - e^{-x}} = i\,\csc\,ix. \!</math>