Complex Exponents and Complex Powers


5.3  Complex Exponents

    In Section 1.5 we indicated that it is possible to make sense out of expressions such as [Graphics:Images/ComplexFunComplexPowerMod_gr_1.gif] or [Graphics:Images/ComplexFunComplexPowerMod_gr_2.gif] without appealing to a number system beyond the framework of complex numbers.  We now show how this is done by taking note of some rudimentary properties of the complex exponential and logarithm, and then using our imagination.

    We begin by generalizing Identity (5-15).  Equations (5-12) and (5-14) show that [Graphics:Images/ComplexFunComplexPowerMod_gr_3.gif] can be expressed as the set [Graphics:Images/ComplexFunComplexPowerMod_gr_4.gif].  We can easily show (left as an exercise) that, for [Graphics:Images/ComplexFunComplexPowerMod_gr_5.gif], [Graphics:Images/ComplexFunComplexPowerMod_gr_6.gif], where [Graphics:Images/ComplexFunComplexPowerMod_gr_7.gif] is any branch of the function [Graphics:Images/ComplexFunComplexPowerMod_gr_8.gif].  But this means that for any [Graphics:Images/ComplexFunComplexPowerMod_gr_9.gif], the identity [Graphics:Images/ComplexFunComplexPowerMod_gr_10.gif] holds true.  Because [Graphics:Images/ComplexFunComplexPowerMod_gr_11.gif] denotes the set [Graphics:Images/ComplexFunComplexPowerMod_gr_12.gif], we see that [Graphics:Images/ComplexFunComplexPowerMod_gr_13.gif], for [Graphics:Images/ComplexFunComplexPowerMod_gr_14.gif].

    Next, we note that identity (5-17) gives us [Graphics:Images/ComplexFunComplexPowerMod_gr_15.gif], where n is any natural number, so that [Graphics:Images/ComplexFunComplexPowerMod_gr_16.gif] for [Graphics:Images/ComplexFunComplexPowerMod_gr_17.gif].  With these preliminaries out of the way, we can now come up with a definition of a complex number raised to a complex power.


Definition 5.4.  Let c be a complex number.  We define [Graphics:Images/ComplexFunComplexPowerMod_gr_18.gif] as follows  

(5-21)            [Graphics:Images/ComplexFunComplexPowerMod_gr_19.gif].  

     The right side of Equation (5-21) is a set.  This definition makes sense because, if both  z  and  c  are real numbers with  [Graphics:Images/ComplexFunComplexPowerMod_gr_20.gif], Equation (5-21) gives the familiar (real) definition for  [Graphics:Images/ComplexFunComplexPowerMod_gr_21.gif], as the following example illustrates.


Example 5.6.  Use Equation (5-21) to evaluate [Graphics:Images/ComplexFunComplexPowerMod_gr_22.gif].

Solution.  Calculating  [Graphics:Images/ComplexFunComplexPowerMod_gr_23.gif]  gives  


Thus  [Graphics:Images/ComplexFunComplexPowerMod_gr_25.gif]  is the set  [Graphics:Images/ComplexFunComplexPowerMod_gr_26.gif].  The distinct values occur when [Graphics:Images/ComplexFunComplexPowerMod_gr_27.gif];  we get  [Graphics:Images/ComplexFunComplexPowerMod_gr_28.gif]  and  [Graphics:Images/ComplexFunComplexPowerMod_gr_29.gif].  In other words, [Graphics:Images/ComplexFunComplexPowerMod_gr_30.gif].  

Remark.  The expression  [Graphics:Images/ComplexFunComplexPowerMod_gr_31.gif]  is different from  [Graphics:Images/ComplexFunComplexPowerMod_gr_32.gif],  as the former represents the set  [Graphics:Images/ComplexFunComplexPowerMod_gr_33.gif]  and the latter gives only one value,  [Graphics:Images/ComplexFunComplexPowerMod_gr_34.gif].

Explore Solution 5.6.


    Because [Graphics:Images/ComplexFunComplexPowerMod_gr_39.gif] is multivalued, the function [Graphics:Images/ComplexFunComplexPowerMod_gr_40.gif] will, in general, be multivalued.  If we want to focus on a single value for [Graphics:Images/ComplexFunComplexPowerMod_gr_41.gif], we can do so via the function defined for [Graphics:Images/ComplexFunComplexPowerMod_gr_42.gif] by

(5-22)            [Graphics:Images/ComplexFunComplexPowerMod_gr_43.gif]  

which is called the principal branch of the multivalued function  [Graphics:Images/ComplexFunComplexPowerMod_gr_44.gif].  Note that the principal branch of  [Graphics:Images/ComplexFunComplexPowerMod_gr_45.gif]  is obtained from Equation (5-21) by replacing  [Graphics:Images/ComplexFunComplexPowerMod_gr_46.gif]  with the principal branch of the logarithm.


Example 5.7. Find the principal value of  (a)   [Graphics:Images/ComplexFunComplexPowerMod_gr_47.gif],  and  (b)  [Graphics:Images/ComplexFunComplexPowerMod_gr_48.gif].  

Solution. From Example 5.3,  


Identity (5-22) yields the principal values of [Graphics:Images/ComplexFunComplexPowerMod_gr_50.gif] and [Graphics:Images/ComplexFunComplexPowerMod_gr_51.gif]:  


Note that the result of raising a complex number to a complex power may be a real number in a nontrivial way.  

Explore Solution 5.7 (a).

Explore Solution 5.7 (b).



                                                             Figure 5.C  The mapping  [Graphics:Images/ComplexFunComplexPowerMod_gr_84.gif].


Complex Exponents

    Let us now consider the various possibilities that may arise in the definition of  [Graphics:Images/ComplexFunComplexPowerMod_gr_85.gif].  

Case (i).  Suppose [Graphics:Images/ComplexFunComplexPowerMod_gr_86.gif] where k an integer.  Then, if  [Graphics:Images/ComplexFunComplexPowerMod_gr_87.gif],  


Recalling that the complex exponential function has period [Graphics:Images/ComplexFunComplexPowerMod_gr_89.gif], we have  


which is the single-valued kth power of z that we discussed in Section 1.5.  This is easily verified by the computation


which is the single-valued kth power of z that we discussed in Section 1.5.


Case (ii).  If  [Graphics:Images/ComplexFunComplexPowerMod_gr_92.gif]  where k is an integer and  [Graphics:Images/ComplexFunComplexPowerMod_gr_93.gif],  then

(5-23)            [Graphics:Images/ComplexFunComplexPowerMod_gr_94.gif].  

Hence Equation (5-21) becomes  

            [Graphics:Images/ComplexFunComplexPowerMod_gr_95.gif]  for  [Graphics:Images/ComplexFunComplexPowerMod_gr_96.gif].

When we again use the periodicity of the complex exponential function, Equation (5-23) gives k distinct values corresponding to  [Graphics:Images/ComplexFunComplexPowerMod_gr_97.gif].  Therefore, as Example 5.6 illustrated, the fractional power [Graphics:Images/ComplexFunComplexPowerMod_gr_98.gif] is the multivalued [Graphics:Images/ComplexFunComplexPowerMod_gr_99.gif] root function.  Equation (5-23) is easily verified by the computation  



Case (iii).  If j and k are positive integers that have no common factors and  [Graphics:Images/ComplexFunComplexPowerMod_gr_101.gif],  then Equation (5-21) becomes  

            [Graphics:Images/ComplexFunComplexPowerMod_gr_102.gif]  for  [Graphics:Images/ComplexFunComplexPowerMod_gr_103.gif].

This is easy to establish.  If  [Graphics:Images/ComplexFunComplexPowerMod_gr_104.gif]  then  


and again there are k distinct values corresponding to  [Graphics:Images/ComplexFunComplexPowerMod_gr_97.gif].  


Case (iv).  Suppose c is not a rational number, then there are infinitely many values for  [Graphics:Images/ComplexFunComplexPowerMod_gr_106.gif],  provided  [Graphics:Images/ComplexFunComplexPowerMod_gr_107.gif].


Example 5.8.  The values of of  [Graphics:Images/ComplexFunComplexPowerMod_gr_108.gif]  are  


where n is an integer.  The principal value of  [Graphics:Images/ComplexFunComplexPowerMod_gr_110.gif]  is  


Figure 5.6 shows the terms for this multivalued expression corresponding to  [Graphics:Images/ComplexFunComplexPowerMod_gr_112.gif].  They exhibit a spiral pattern that is often present in complex powers.

Explore Solution 5.8.



    Figure 5.6  Some of the "spiral pattern" of values for  [Graphics:Images/ComplexFunComplexPowerMod_gr_130.gif].


The Rules for exponents.  

    Some of the rules for exponents carry over from the real case.  In the exercises we ask you to show that if c and d are complex numbers and  [Graphics:Images/ComplexFunComplexPowerMod_gr_131.gif],  then  

where n is an integer.  

    The following example shows that Identity (5-21) does not hold if n is replaced with an arbitrary complex value.


Example 5.9.  (a)  [Graphics:Images/ComplexFunComplexPowerMod_gr_133.gif],  

and  (b) [Graphics:Images/ComplexFunComplexPowerMod_gr_134.gif][Graphics:Images/ComplexFunComplexPowerMod_gr_135.gif].  

Since these sets of solutions are not equal, Identity (5-21)  does not always hold.

Explore Solution 5.9.


    We can compute the derivative of the principal branch of [Graphics:Images/ComplexFunComplexPowerMod_gr_145.gif], which is the function  

(5-28)            [Graphics:Images/ComplexFunComplexPowerMod_gr_146.gif].   By the chain rule,  
(5-28)            .  

    If we restrict [Graphics:Images/ComplexFunComplexPowerMod_gr_148.gif] to the principal branch,  [Graphics:Images/ComplexFunComplexPowerMod_gr_149.gif], then Equation (5-28) can be written in the familiar form that you learned in calculus.  That is, for [Graphics:Images/ComplexFunComplexPowerMod_gr_150.gif] and z not a negative real number,




    We can use Identity (5-21) to define the exponential function with base b, where [Graphics:Images/ComplexFunComplexPowerMod_gr_154.gif] is a complex number:


    If we specify a branch of the logarithm, then [Graphics:Images/ComplexFunComplexPowerMod_gr_156.gif] will be single-valued and we can use the rules of differentiation to show that the resulting branch of [Graphics:Images/ComplexFunComplexPowerMod_gr_157.gif] is an analytic function.  The derivative of [Graphics:Images/ComplexFunComplexPowerMod_gr_158.gif] is then given by the familiar rule  

(5-29)            [Graphics:Images/ComplexFunComplexPowerMod_gr_159.gif]

where [Graphics:Images/ComplexFunComplexPowerMod_gr_160.gif] is any branch of the logarithm whose branch cut does not include the point [Graphics:Images/ComplexFunComplexPowerMod_gr_161.gif].


Exercises for Section 5.3.  Complex Exponents


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Exponential Function

Complex Logarithms


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