1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

184 CHAPTER 5 • ELEMENTARY FUNCTIONS

The derivatives of the hyperbolic functions follow the same rules as in cal-
culus:

! coshz = sinhz and

~tanhz = sech^2 z and

dz

d

dz sechz = - sech z tanh z

:z sinhz = coshz;

~ cothz = -csch^2 z·

dz '

d

and dz cschz = -cschz coth z.

The hyperbolic cosine and hyperbolic sine can be expressed as

cosh z = cosh x cosy + i sinh x sin y and

sinhz = sinhxcosy + icoshxsiny.

The complex trigonometric and hyperbolic functions are all defined in terms

of the exponential function, so we can easily show them to be related by

cosh(iz)=cosz and sinh (iz)=isinz,

sin(iz) = isinhz and cos(iz) = coshz.

Some of the important identities involving the hyperbolic functions are

cosh^2 z -sinh^2 z = 1,

sinh(z 1 +z2) = sinhz 1 coshz2 +coshz1sinhz2,

cosh (z 1 + z2) = coshz1 coshz2 + sinhz1 sinhz2,

cosh (z + 2?ri) = cosh z,

sinh (z + 2?ri) = sinh z,

cosh(-z) = coshz, and

sinh(-z) = - sinhz.

We conclude this section with an example from electronics. In the theory of

electric circuits, the voltage drop, ER, across a resistance R obeys Ohm's law,

or

En=IR,

where I is the current flowing through the resistor. Additionally, the current

and voltage drop across an inductor, L , obey the equation

dl

Ei=L-.

dt

The current and voltage across a capacitor, C, are related by

Ec = c^1 lt I(r)dr.

to