Calculus: Analytic Geometry and Calculus, with Vectors

(^506) Exponential and logarithmic functions
and hence that
(9.312) eu = cos z + i sin z
(9.313) e u = cos z - i sin z.
Adding and subtracting give
(9.314) eiz + e-iz

ei. - e"

2 sin z = 2i

These are Euler (1707-1783) formulas. They are widely used, particu-

larly in electrical engineering, to replace calculations involving sines,

cosines, and their positive powers by simpler calculations involving expo-

nential,. We do not need to understand these matters now, but we can

at least acquire a vague feeling that trigonometric functions are related

to exponential functions by formulas very similar to those which relate

hyperbolic functions to exponential functions. In any case, we can know

that there is a reason why formulas involving hyperbolic functions are

so similar to formulas involving trigonometric functions. Modern

scientists know that their ancestors complicated many problems by

habitually using trigonometric and hyperbolic functions in situations in

which results are obtained much more neatly and quickly by use of

exponentials. Thus hyperbolic functions are introduced to students

with the hope that they will (insofar as they can control their own

activities) use the functions only for purposes for which they are useful.

We now return to our usual situation in which all numbers are real.

The hyperbolic sine, hyperbolic cosine, and other hyperbolic functions are

defined by the first of the following equations, and calculation of the

derivatives gives practice in differentiation.

z

(9.32) sinh x =ex - e- 2 , dx sinh x = cosh x

(9.321) cosh x =ex

2

e

dx

cosh x = sinh x

(9.322) tank x = ez+e_z, dtank x = sech2 x

(9.323) coth x = ex + e-, d coth x = - csch2 x

(9.324) sech x =

ex

+

e

s, dsech x = - sech x tank x

(9.325) csch x = 2

e-y, d csch x = - csch x coth x

These and many other formulas are very similar to trigonometric for-

mulas, but differences in signs must be noted. As is the case for trigo-