5.5  Inverse Trigonometric and Hyperbolic Functions

    We expressed trigonometric and hyperbolic functions in Section 5.4 in terms of the exponential function.  In this section we look at their inverses.  When we solve equations such as    for z,  we will obtain formulas that involve the logarithm.  Because trigonometric and hyperbolic functions are all periodic, they are many-to-one; hence their inverses are necessarily multivalued. The formulas for the inverse trigonometric functions are

     

    We derive the first equation and leave the others as exercises.  If we take a particular branch of the multivalued function, , we have  

            

which we can also write as  

            .  

    Multiplying both sides of this equation by    gives  
    
            ,  

which is a quadratic equation in terms of  .  Using the quadratic equation to solve for  ,  we obtain  

            ,  

where the square root is a multivalued function.  Taking the logarithm of both sides and obtain  ,  then simplify this and obtain the desired result:

            ,

where the multivalued logarithm is used.  To construct a specific branch of , we must first select a branch of the square root and then select a branch of the logarithm.

Remark.  The function    is known to be an odd function and therefore  .  As an immediate consequence we have

            .

We can now use the trigonometric identity    and obtain an alternate expression for :  

               

We mention this derivation because Mathematica uses the second line for determining the principal value, i.e.

            

an we will see shortly that  .

The principal value used by Mathematica for the inverse trigonometric functions are

                

Exploration

    When the principal value is used,    maps the upper half-plane    onto a portion of the upper half-plane     that lies in the vertical strip  .  The image of a rectangular grid in the z plane is a "spider web" in the w plane, as Figure 5.10 shows.

        Figure 5.10  A rectangular grid is mapped onto a spider web by  .

The derivatives of   arcsin(z), arccos(z), arctan(z)   

    We can find the derivatives of any branch of these functions by using the chain rule:
    
  

Exploration.

    We get the derivative of    by starting with the equation    and differentiating both sides, using the chain rule:

                

Then    is used to get the substitution for the preceding equation, and we obtain the desired result:

            .

Derivation of the other formulas are left as exercises for the reader.

The Inverse Sine  arcsin(z) .   Verify that the formula  

(i)               

is correct.  (At least for values of z in the upper half plane  .)

Explore Formula (i).

The Inverse Cosine  arccos(z) .   Verify that the formula(s)   

(ii a)            ,  

(ii a)            .  

are correct.  (At least for values of z in the upper half plane  .)

Explore Formula (ii a).

Explore Formula (ii b).

The Inverse Tangent  arctan(z) .  Verify that the formula  

(iii)              

is correct.  (At least for values of z in the upper half plane  .)

Explore Formula (iii).

    When the principal value is used,    maps the upper half-plane    onto a portion of the upper half-plane     that lies in the vertical strip  .  

                Figure 5.B  The mapping  .

Example 5.11.  The values of    are given by  

            .  

Using straightforward techniques, we simplify this equation and obtain  



where n is an integer.  We observe that  

            
and then write

            ,   where n is an integer.

Explore Solution 5.11.

Example 5.12.  Suppose that we make specific choices in equation (5-47) by selecting    as the value of the square root    and using the principal value of the logarithm.  With  ,  the result is  

            ,  

and the corresponding value of the derivative is given by

            .  

Explore Solution 5.12.

The Inverse Hyperbolic Functions

    The inverse hyperbolic functions are  

   

    To establish the last identity given above, we start with    and obtain  

            

which we solve for  ,  getting  .  Taking the logarithms of both sides gives  

            ,

which is what we wanted to show.

The Inverse Hyperbolic Sine  arcsinh(z).  Verify that the formula  

(iv)              

is correct, we can verify this graphically.  (At least for values of z in the upper half plane first quadrant.)

Explore Formula (iv).

The Inverse Hyperbolic Cosine  arccosh(z) .  Verify that the formula  

(v)              

is correct, we can verify this graphically.   But it is correct only in quadrants  I  and  IV.

Explore Formula (v).

The Inverse Hyperbolic Tangent  arctanh(z) .  Verify that the formula  

(vi)               
            
is correct, we can verify this graphically.

Explore Formula (vi).

        The derivatives of the inverse hyperbolic functions are

              

Exploration.

Example 5.13.  Calculation reveals that  

              

where n is an integer.

Explore Solution 5.13.

Library Research Experience for Undergraduates

Exponential Function

Complex Logarithms

Conformal Mapping

Smith Chart

Download this Mathematica Notebook

Return to the Complex Analysis Project

(c) 2006 John H. Mathews, Russell W. Howell