5.1 The Complex Exponential Function

    Recall that the real exponential function can be represented by the power series  [Graphics:Images/ComplexFunExponentialMod_gr_4.gif].  Thus it is only natural to define the complex exponential  [Graphics:Images/ComplexFunExponentialMod_gr_5.gif],  also written as [Graphics:Images/ComplexFunExponentialMod_gr_6.gif], in the following way:


Definition 5.1 (Exponential Function).  The definition of  exp(z) is  

            [Graphics:Images/ComplexFunExponentialMod_gr_7.gif].  

Demonstration for Definition 5.1.  

We can use Mathematica to sum a power series.

[Graphics:../Images/ComplexFunExponentialMod_gr_8.gif]


[Graphics:../Images/ComplexFunExponentialMod_gr_9.gif]
[Graphics:../Images/ComplexFunExponentialMod_gr_10.gif]


The ratio test can be used to show that the infinite series converges for all values of   z.

[Graphics:../Images/ComplexFunExponentialMod_gr_11.gif]




[Graphics:../Images/ComplexFunExponentialMod_gr_12.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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