Mathematical Tools for Physics

1—Basic Stuff 3

The other three functions are

sechx=

1

coshx

, cschx=

1

sinhx

, cothx=

1

tanhx

Drawing these is left to problem 4 , with a stopover in section1.8of this chapter.

Just as with the circular functions there are a bunch of identities relating these functions. For the analog
ofcos^2 θ+ sin^2 θ= 1you have

cosh^2 θ−sinh^2 θ= 1 (2)

For a proof, simply substitute the definitions ofcoshandsinhin terms of exponentials. Similarly the other
common trig identities have their counterpart here.

1 + tan^2 θ= sec^2 θ has the analog 1 −tanh^2 θ= sech^2 θ (3)

The reason for this close parallel lies in the complex plane, becausecos(ix) = coshxandsin(ix) =isinhx. See
chapter three.

The inverse hyperbolic functions are easier to evaluate than are the corresponding circular functions. I’ll
solve for the inverse hyperbolic sine as an example

y= sinhx means x= sinh−^1 y, y=

ex−e−x

2

Multiply by 2 exto get the quadratic equation

2 exy=e^2 x− 1 or

(

ex

) 2

− 2 y

(

ex

)

−1 = 0