csch

The following formula holds for all real numbers x except 0: csch À x ¼ Àcsch x HYPERBOLIC SECANT OF NEGATIVE VARIABLE The hyperbolic secant of the negative of a variable is equal to the hyperbolic secant of the variable.

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SOLUTION 4-1 Remember that the hyperbolic cosecant (csch) is the reciprocal of the hyper - bolic sine (sinh).

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Thus, csch x, which is equal to 1/(sinh x) and is graphed in Fig. 4-4, grows without limit as x!

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Relations to Trigonometric Functions sinh (z) = - i sin (iz) csch (z) = i csc (iz) cosh (z) = cos (iz) sech (z) = sec (iz)

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THE OTHER FOUR The remaining four hyperbolic functions follow from the hyperbolic sine and the hyperbolic cosine, like this: tanh x ¼ sinh x = cosh x csch x ¼ 1 = sinh x sech x ¼ 1 = cosh x coth x ¼ cosh x = sinh x In terms of exponential functions, they are expressed this way: tanh x ¼ ðex À eÀx Þ = ðex þ eÀx Þ csch x ¼ 2 = ðex À eÀx

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