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Euler’s Formula by Series Expansions by waabu

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This paper verifies Euler’s Formula by series expansions and graphical methods at an elementary level. Binomial and Exponential series expansions are used to show a means of proof alternative to the classical Maclaurin Series. Wikipedia has fairly good article titled "Euler's Formula". It shows other ways to prove exp(ix) & exp(-ix).

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Euler’s Formula by Series Expansions
╬

Francis J. O’Brien, Jr., Ph.D.
Aquidneck Indian Council
Newport, RI

 Francis   J. O’Brien, Jr.                                                               July 4, 2013

On complex numbers and Euler’s Formula—

“It is one of the truly amazing facts of mathematics that the
use of complex numbers simplifies many problems, from the
convergence of series to the evaluation of definite integrals,
which on their face seem to belong strictly to the real
domain.” (Gleason, p.129)

▲

“Gentlemen, that is surely true, it is absolutely paradoxical;
we cannot understand it, and we don't know what it means.
But we have proved it, and therefore we know it must be the
truth”. Benjamin Pierce (1809-1880)

Introduction

This paper verifies Euler’s Formula, e ix  cos x  i sin x, by series expansions and
graphical methods at an elementary level. Binomial and Exponential series expansions are
used to show a means of proof alternative to the classical Maclaurin Series1. The justification
for using real only terms in differentiation and integration of circular functions is substantiated
by analyzing the real parts only of the series.
A basic review and summary is provided for the Binomial Theorem, Binomial &
Exponential Series, and Maclaurin Series before complex numbers and the derivations.

1                                                                                                    
Since the Binomial Series is derived by Maclaurin Series calculations on the function f ( x )  a  x this
n

approach may seem like circular reasoning. But the intention is to make the paper accessible to readers not
necessarily familiar with infinite series. Experience teaches that students readily accept the Binomial Series as a
plausible extension of the integer Binomial Theorem (which requires only algebra and Mathematical Induction for
justification).
© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 2 of 16

Binomial Theorem for Integer Exponents
n   n
( a  b) n      k a k b n  k ,
                       n  1,2, 
k  0 
n        n!        n(n  1)(n  2)  n  k  1
 
 k  k! (n  k )!                                k  1, 0!  1!  1.
                            1  2  3 k

n(n  1) n  2 2
(a  b) n  a n  na n 1b                a     b  ...  nab n 1  b n               (written in reverse order)
2!

NOTE: We can also express (a  b) n in an equivalent form by interchanging the exponents:

n   n
( a  b)    n
  k a n  k b k
 
k  0   

n
 b
NOTE: In some calculations it is easier to use the form, (a  b) n  a n 1   .
 a

Binomial Series for Real Numbers

 n  k nk     n
( a  b)    
n
 a b         a n  k b k
k 
k
k  0            k  0 

NOTE: n is any real number not necessarily a positive integer. The Binomial Series may be
viewed as an unending version of the Binomial Theorem when n is noninteger. It is very
useful for theoretical analysis of the behavior of binomial limits such as the exponential
function used in this paper2.

Letting a  1,

   n        n       n         n             n        n     n
(1  b) n      k b k   0 b0  1 b1   2 b 2   3 b3   4 b 4   5 b5  
                                                   
k  0                                                         

x n
2The well-known limit definition is: e x  lim  1  . Verifiable with L’Hôpital’s Rule by taking log. of limit.
     
n   n 
© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 3 of 16

n
Substituting   values,
k 
 
(1  b) n
b2                   b3                          b4
 1  nb  n(n  1)       n(n  1)(n  2)     n(n  1)(n  2)(n  3)
2!                   3!                          4!
b5                                        b6                 Binomial
 n(n  1)(n  2)(n  3)(n  4)     n(n  1)(n  2)(n  3)(n  4)(n  5)
5!                                        6!                 Series
b7
 n(n  1)(n  2)(n  3)(n  4)(n  5)(n  6)    
7!

n
NOTE: This series converges if b  1. See Bers (p. 540) or O’Brien, “Derivative of e x by
the Limit Definition”, 2013.

Maclaurin Series Expansion for Approximating a Function                                     f (x)

Definition: A power series for infinitely many derivatives (Finney & Thomas).

              k
x2          x3             xn           k (0) x
f ( x)  f (0)  f (0) x  f 0   f (0)           (n)
   f (0)       f
2!          3!             n!      k 0
k!

where f (0) is the first derivative evaluated at 0, f 0  is the second derivative evaluated at 0,
etc.

      Example: Expansions of Cosine & Sine

d
cos( x)  f ( x)   sin x    cos(0)  1
dx
d
sin( x)  f ( x)  cos x       sin(0)  0
dx

x2 x4 x6      1n x 2n      1k x 2k
cos x  1        
2! 4! 6!

2n !            2k !
k 0

x3 x5 x7       1n x 2n 1      1k x 2k 1
sin x  x        
3! 5! 7!

2n  1!              2k  1!
k 0

© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 4 of 16

NOTE: We will show alternative series expansions for Euler’s Formula later in the paper
using these classical results for cos and sin as verification.

Exponential Series

We now derive with a fair amount of mathematical detail the series expansion of e x
from the definition of the limit of e x and the binomial series expansion. It is used to obtain
Euler’s Formula.

The limit definition of e x is:

n
x              x                                                x                   
e  lim 1                                                 Set b  in Binomial Series 
n           n                                                n                   

Expanding e x by the Binomial Series (see next page):

© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 5 of 16

 n
n                      k
   x                  x
e  lim 1    lim    
x
k  n
n     n  n   k  0  

 n  x  0  n  x 1  n  x  2  n  x 3  n  x  4  n  x 5    
 lim                         
           1  n                  3  n                 5  n
n    0  n        2  n              4  n                   
                                                                              
       x  n(n  1)  x 
2
n(n  1)(n  2)  x 
3
n(n  1)(n  2)(n  3)  x  
4
1  n                                                                   
      n        2!  n              3!        n               4!           n 
 lim                                                                                        
n                                           5
 n(n  1)(n  2)(n  3)(n  4)  x   
                                                 

                  5!               n                                                

       n  n  1  x 2 n  n  1  n  2  x 3 n  n  1  n  2  n  3  x 4 
1  x                                                          
       n  n  2! n  n  n  3! n  n  n  n  4! 
 lim
n    n  n  1  n  2  n  3  n  4  x 5                                       
                                                                       
 n  n  n  n  n  5!
                                                                                  

x2       1 x
3
 1        2
 lim 1  lim x                   lim 1      lim 1   lim 1  
n        n        2! n   n  3! n   n  n   n 
x4       1        2        3
       lim 1   lim 1   lim 1  
4! n   n  n   n  n   n 
x5       1        2        3        4
       lim 1   lim 1   lim 1   lim 1    
5! n   n  n   n  n   n  n   n 

 k
x 2 x3 x 4 x5              x
1 x              ...                                           Exponential
2! 3! 4! 5!           k 0
k!                                     Series

_____________

This is the primitive infinite series of e x x  0 derived from the binomial expansion
based on the limit definition of e x . The expansion for e x is easily derived with the same result
by the Maclaurin Series since any derivative of e x is always e x as any calculus texts shows3.
The exponential series converges for all values of x by the Ratio Test.

NOTE: if we set x   x the negative exponential series is,

3   One must always check that the constant term f (0) in the Maclaurin Series is included; in our case
0
f (0)  e  1 is provided in the expansion.
© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 6 of 16

   x
n
e  x  lim 1    1  x 
x 2 x3 x 4 x5
         ...  

 1k x k
n     n           2! 3! 4! 5!           k 0
k!

In general, for any differentiable function g (x), the exponential series is:


g ( x)k
e g ( x)             k!
k 0

For example, if we set x  ax in Exponential Series,

 ax 
n
ax 2  ax 3  ax 4  ax 5  ...   ax k
e   ax
 lim 1    1  ax 
n     n              2!       3!       4!       5!
 k!
k 0

Similarly, if we set x  ax in Exponential Series,

 ax 
n
e  ax  lim 1    1  ax 
ax 2  ax 3  ax 4  ax 5  ...    1k ax k
n     n              2!       3!       4!       5!
 k!
k 0

______________

Complex Numbers4
The formula we seek to verify is:

     Euler’s Formula

eix  cos x  i sin x
x is real
e  ix  cos x  i sin x

NOTE: For verification of e ix , recall the relations,

cos x   cosx 
sin  x    sin x 

They follow by setting x   x in the cosine & sine Maclaurin Series expansions above.

4A good introductory treatment is found in Boas (Ch. 2) and an advanced treatment in Gleason (Ch. 10 & Ch. 15).
© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 7 of 16

NOTE: Euler’s Formula is justified typically by Maclaurin series expansions on e ix (e.g.,
Finney and Thomas, Ch. 9; Boas, Ch. 2; Gleason, Ch.15). We will show a different proof
below.

NOTE: To isolate cos x from Euler’s Formula add each term and subtract each term for sin:

eix  e ix
cos x 
2
eix  e ix
sin x 
2i

This is the formal definition of cosine & sine. For real parts only the following is used.

Euler’s Identity & Complex Exponentials

These relations establish important consequences of Euler’s Formula useful for
calculations by complex exponentials.

i
ei  1       [Identity]                                                        i 2  1
e2          112
i
k                                                       k
 k        k                      i
2  i k  cos k   i sin  k 
i
e 2  i k  cos   i sin                      e                                     
 2          2                                          2        2 
NOTE: to verify, substitute all into Euler’s Formula; e.g., e i  cos  i sin    1.

NOTE: It is clear that the real parts of Euler’s Formula are:

o        Re eix  Re(cos x  i sin x)  cos x [by inspection and definition of “real”]
o        Re e ix  Re(cos x  i sin x)  cos x
ix
 cos x  i sin x       e
o        Re                   Re          Re  ie ix  sin x
       i                i
ix
 cos x  i sin x        e
o        Re                   Re             Re ie  ix  sin x
       i                 i

NOTE: These forms can also be established by series expansions. The reader can also verify
from these definitions that cos x   cosx & sin  x    sin x .

________________

© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 8 of 16

Graphical Representation of Euler’s Formula

Figure 1 is a plot of Euler’s Formula which graphs the real part and imaginary part for each
value of x (Argand diagram). The real part is the cosine function, as asserted above (what
function is the imaginary part?).

NOTE: Example of the plot values of eix in Figure 1:

ei  cos( )  i sin    1,0

ei 2.25  cos(2.25)  i sin 2.25  .63,.78i 
   The x axis is real; the y axis is imaginary
   The values of cos and sin will be equal on the graph by the following relations
     
cos  x   sin( x)
2     
     
sin   x   cos( x)
2     

The plot values for e ix are similar.
eix
Fig. 2 is a plot of   & ie  ix . The real parts are the sine function, as asserted above (what
i
function is the imaginary part?). Analysis below will show justification.

© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 9 of 16

e ix

e ix

Figure 1. Plot of eix & e ix . Real parts are the cosine function.
Source: http://www.wolframalpha.com

© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 10 of 16

e ix
i

ie ix

eix
Figure 2. Plot of     & ie  ix . Real parts are the sine function.
i
Source: http://www.wolframalpha.com

__________________

© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 11 of 16

_________

The powers of the imaginary number i are displayed in Table 1. Any odd/even value of
k
i can be written down with this structure, useful for writing the individual terms of
expansions presented next.

Table 1
Powers of Imaginary Number
i 2  1
k       ik               Value of i k            Re i k
0            i0                          1              X
1             i1                       i
2            i2                            –1            X
3             i3                    –i
4            i4                            1             X
5             i5                       i
6            i6                            –1            X
7            i7                     –i
8             i8                           1             X
9             i9                       i
10           i10                           –1            X
11           i11                  –i

k                          1 if k 2 even

k
(even)      i 2 2                                       X
 1 if k 2 odd
k                k 1
i if k  1 2 even
(odd)

i i2 2           
 i if k  1 2 odd

© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 12 of 16

Series Expansion of Euler’s Formula e ix
We now show by series expansions the justification of Euler’s Formula. The real parts
only calculations of Euler’s Formula for cosine and sine are based on the definitions given
earlier.

By the limit definition of eix and Exponential Series5:

    e   ix          ix 
n
 lim 1    
ix 
        k

n     n   k 0
k!

 1  ix 
ix 2  ix 3  ix 4  ix 5  ix 6  ix 7   
2!     3!       4!         5!       6!      7!
x2    x3 x 4    x5 x6   x7
 1  ix        i       i      i 
2!    3! 4!     5! 6!   7!
 x
cos
                               sin x 

           
2     4     6                            3     5     7                            set x  ix in
x     x     x                             x     x     x
1                               i x                                          Exponential Series
2! 4! 6!                                 3! 5! 7!           
                                                use Table 1 i k values
 cos x  i sin x                                                                         isolate real & imag.
terms
    compare with Maclaurin
ix                                                      Series to verify
By the limit definition of e             and Exponential Series:


 ix 
    n
e  ix  lim 1      1k
ix k
n     n   k 0
k!

The calculation will show,

e ix  cos x  i sin x.

Thus, we have demonstrated by series expansions, e ix  cos x  i sin x

5 In this calculation x is treated as a real variable and i is a number with properties of Table 1. This calculation is
 ax
the one given above for e ax with i playing the role of a. The same applies to e         .
© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 13 of 16

    Real Parts Only Analysis

       ix  
n        ix k 
 cos x  Re e      ix
 Re  lim 1     Re          
 n    n        k  0 k! 
          
              


 Re 1  ix 
ix 2  ix 3  ix 4  ix 5  ix 6  ...


              2!       3!       4!       5!       6!         

          x2      x3 x 4         x5 x6          
 Re 1  ix         i           i             

          2!       3! 4!          5! 6!         

x2 x4 x6
1              cos x
2! 4! 6!                                                                                 set x  ix in
Exponential Series
eix        1        ix  
n         1  ix k                   use Table 1 i k values
 sin x  Re            Re  lim 1     Re                    
i         i n       n           i k  0 k!                  isolate real only
                                                        terms
           ix 2  ix 3  ix 4  ix 5  ix 6  ix 7  ...                     compare with
1  ix                                                                                 Maclaurin Series to
 Re              2!       3!       4!       5!       6!       7!         
                                  i                                                       verify
                                                                                     
                                                                     
         x2    x3 x 4    x5 x6    x7    
1  ix     i       i      i       
 Re          2!    3! 4!     5! 6!    7!  
                   i                    
                                        
                                        
x3 x5 x7
 x             sin x
3! 5! 7!

NOTE: substituting  x for x in each series shows:

cos( x)  cos( x)
sin( x)   sin( x)

© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 14 of 16

       ix                k ix 
ix                ix
n                   k
NOTE: for the real part of e expand Re e         Re  lim 1     Re   (1)           ,
 n    n       k  0
            k!  
               
         ix                k ix 
ix            ix
n                    k
and for the real part of ie expand Re ie       Re i lim 1     Re i  (1)          .
 n      n       k 0
             k!  
                
The results will demonstrate that cos x  Re e ix & sin x  Re ie ix . These results justify
integration using real parts only6.
sin ax
For example, the indefinite integral  cos(ax)dx          (ignoring the constant of
a
integration) can be done using complex exponentials in two ways. By the classical definition
of Euler’s Formula,

 eiax  e iax         iax     iax 
               dx  1  e  e          sin( ax) .
 cos(ax)dx          2            a     2i             a
                                     

 cos(ax)dx  Re           dx 
iax
For real parts analysis we can choose                           e

 1  cos(ax)  i sin(ax)  sin(ax)
1  eiax
Re     Re
          , or
a  i   a            i                 a
   

                              1  e  iax 
cos(ax)dx  Re  e  iax dx   Re
a  i 
   1 Re  cos(ax)  i sin(ax)   sin(ax) .
a            i           
      a


NOTE: In a similar manner differentiation of cosine and sine can be derived by real parts
analysis (one chooses eix or e ix depending on the problem to solve; we show all options).


d
dx
d
 
cos( x)  Re eix  Re ie ix  Re i cos x  i sin x    sin x 
dx

 Reix  Re cos x  i sin x   cosx 
d               d eix
        sin( x)  Re           e 
dx              dx i
d               d
        cos( x)  Re e  ix   sin x 
dx             dx
d              d
        sin( x)  Re ie  ix  cosx 
dx             dx

o Now derive the same with the classical definitions of Euler’s Formula for
cosine & sine.

6A good YouTube lecture from MIT on integration of circular functions with complex numbers is at “18.03
Differential Equations” (Lecture 6).
© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 15 of 16

The real parts analysis is easier for differentiation and integration. It is justified by the
results in this paper and Maclaurin series expansions (see Finney and Thomas, p. 634, and for
an advanced treatment, Gleason, p. 310).

The author has several online papers at http://www.docstoc.com/profile/waabu which
uses complex numbers to evaluate difficult circular integrals.   Complex numbers makes it
possible to solve such integrals which computer sometimes cannot evaluate.

A future paper will use complex numbers to verify well-known circular identities, such
as sin 2   cos 2   1, cos   , etc. The derivations are easier than the traditional real-

variable method in trigonometry.

© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.
Page 16 of 16

SOURCES & REFERENCES

Bers, L. Calculus, Vol. 1. Holt, Rinehart and Winston, Inc., 1969.

Boas, Mary L. Mathematical Methods in the Physical Sciences, 3rd ed., 2006. NY: Wiley.

Finney, R.L. and G.B. Thomas, Jr. Calculus, 1990. Addison-Wesley Publishing Co.

Gleason, A. M. Fundamentals of Abstract Analysis. Boston: Jones and Bartlett, 1991.

MIT, “Differential Equations”. http://ocw.mit.edu/courses/mathematics/18-03-
differential-equations-spring-2010/index.htm

n
O’Brien, F. J., Jr. “Derivative                     of     e x by      the     Limit     Definition,”   2013
http://www.docstoc.com/profile/waabu.

_______. “A General Limit for Exponentials,” 2013. http://www.docstoc.com/profile/waabu.

Spiegel, M. R. Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill,
1968. [Reprinted Spiegel, Murray R., John Liu, and Seymour Lipschutz. Mathematical
Handbook of Formulas and Tables. New York: McGraw-Hill, 1999.]

© Francis. J. O’Brien, Jr., 2013. <> Aquidneck Indian Council. <> All rights reserved.

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