# Exponential Circular Integrals by waabu

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Demonstrates the evaluation of complicated algebraic-exponential-circular integrals by elementary complex numbers (Euler's Formula, DeMoivre's Theorem) and Gamma Function
NOTE: Wikpedia article "Euler's Formula" can be consulted also.

ERRATA: page 12, next to last step of derivation: put Re before 1/4 and add a bracket.

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Exponential Circular Integrals
╬
Francis J. O’Brien, Jr., Ph.D.
Aquidneck Indian Council
Newport, RI

©Francis J. O’Brien, Jr.                                                                      June 6, 2013

“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)

Introduction

In previous papers complex numbers and the gamma function have been used to
evaluate two classes of integrals,


 dx
cos x n
        
 sin x n
xm 
 

0            

“General Cosine and Sine Integral of Powers,” 2013

and
               cos n bx 
            
 x m 1e  ax             dx,     Re integer n  1
0              sin n bx  
            

“Algebraic–Exponential–Trigonometric Integrals of Integer Powers,” 2013

In this paper we consider additional exponential circular integrals of the form,

                      
m 1  ax  cos bx  cos cx  
                     n         p
                n                        dx,       Re integer n, p   1

x     e
 p
 sin bx   sin    cx  
0                                           

Page 2 of 19

In order to make this paper relatively self-contained, well known results for complex
numbers and circular functions (Spiegel) are summarized.

Elementary Complex Numbers and Exponentials

Familiar elementary complex number definitions and relations are summarized for
reference:

   Euler’s Formula

eix  cos x  i sin x or ei              cos  i sin 
e ix  cos x  i sin x or e i  cos  i sin 

“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)

NOTE: Formulas are derived by Maclaurin series expansions on e ix (Finney and
Thomas, Ch. 9 or Gleason, Ch.15). A future paper will show how to obtain the same
results by an infinite Binomial Series expansion., with tentative title, “Binomial Series
and Euler’s Formula”.

   Euler’s identity
o     ei  1

i
o    e 2   11 2  i
o    i 2  1
m
i           m          m 
o    e 2  i m  cos      i sin    
 2         2 
m
i
o    e            2  i m  cos m   i sin m 
                
 2          2 

   By definition from Euler’s Formula,

Page 3 of 19

eix  e ix
cos x                .
2

o If the cosine power is 1 and one selects e ix or e  ix in Euler’s Formulas for
analysis, one may use real relations for Euler’s Formula
         
Re e ix or Re eix  cos x and
Re eix  or Re e ix    cos x assuming x is real
 Use Re e ix   cos x for convergence of integrals of negative
exponentials when the real only part is desired

   By definition from Euler’s Formula,
eix  e ix
sin x      .
2i
o If the sine power is 1 and one selects e ix or e  ix in Euler’s Formulas for
analysis, one may use real relations for Euler’s Formula

    Re
 e ix 

 i 
                        ix 
 
  Re  ie ix  sin x, and Re  e   Re ie ix   sin x
 i 
                                         
 e  ix 
or Re 
     i 
                      ix 
      
  Re ie  ix  sin x, and Re e   Re  ie  ix   sin x
 i 
                                           
assuming x is real

 e  ix 
 Use Re 
     i 
 
  Re ie  ix  sin x for convergence of
        
integrals of negative exponentials when the real only part is
desired

   Complex conjugation

o    a  bi a  bi    a  bi  a  bi   a 2  b 2

o          m
im  im   i2  1
o
1

1      a  bi   a  bi 
a  bi  a  bi  a  bi  a 2  b 2

Page 4 of 19

NOTE: the real part formulas for cosine and sine are valid only for powers of 1 when x is
1  cos(2 x)
real. For integer powers higher than 1 use identities such as, sin 2 x               , then real
2
parts analysis on unitary power identity. Likewise, for circular products, such as sin A cos B,
reduce by expansion, then real parts analysis on the unitary function. This approach
simplifies calculations and avoids conceptual errors when converting to complex exponential
form1.
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

 eiax  e  iax           iax     iax    sin( ax )
of Euler’s Formula,  cos( ax ) dx                     dx  1  e  e                    .
       2               a      2i             a
                                         
For real parts analysis we can choose  cos(ax)dx  Re  eiax dx  
1  e iax   1  cos( ax )  i sin( ax )  sin( ax )
Re         Re
          , or
a  i       a                 i               a
       

                          
cos(ax)dx  Re  e  iax dx   Re
1  e  iax 
a  i 
   1 Re  cos( ax )  i sin( ax )   sin( ax ) . The
a              i             
      a
      
real parts analysis is easier. It is justified by Maclaurin series expansions (see Finney and
Thomas, p. 634, and for an advanced treatment, Gleason, p. 310).

Circular Functions

For any real variables A & B, the following product relations and identities are stated in real
variable and complex exponential form using Euler’s Formula:

sin( A  B)  sin( A  B)  eiA  e  iA  eiB  e  iB 
sin A cos B                                                          
2                  2i             2       
                            
cos( A  B)  cos( A  B)  e iA  e  iA  eiB  e  iB 
cos A cos B                                                        
2                    2             2       
                             
cos( A  B)  cos( A  B)  eiA  e  iA  eiB  e  iB 
sin A sin B                                                          
2                  2i            2i       
                            

1
A good YouTube lecture from MIT on integration of circular functions with complex numbers is at “18.03
Differential Equations” (Lecture 6).

Page 5 of 19

2
1  cos(2 A)  eiA  e iA 
2
cos A                           
2            2      
             
2
1  cos(2 A)  eiA  e iA 
2
sin A                           
2           2i      
             

1  cos 2 A  cos 2 B cos(2 A  2 B)  cos(2 A  2 B)
sin 2 A cos 2 B                           
4                           8
2                     2
 eiA  e iA              eiB  e iB 
                                      
     2i                        2      
                                      

Gamma Function Reduction Formula

The following algebraic-exponential formula from Gradshteyn & Rhyzik2 (GR) is used to
simplify the integral components:


             m 1
m  x
, Re ,   0, n  0
n
 x e dx           n    
, 
n
0
where    is the gamma function.

For our multivariate integrals set m  m  1,   a, n  1, and use the expression,


m 1  ax            m 
x       e       dx 
am
, m  0, a  0.
0

This is given as Formula 3.326.2 in GR (7th ed.). It is derived in the online paper “ x m e   x dx and related
n

2

integrals, 2nd ed.”, Jan. 3, 2013, http://www.docstoc.com/profile/waabu. See L. Bers, Calculus (pp. 402-403)
for elementary properties of the gamma function and GR (Sect. 8.3) for more extensive information.

Page 6 of 19

Derivation of Integrals

First Degree

We will present three integrals corresponding to each first degree circular product
form. The first is worked out in detail.


m 1
Derivation of       x          exp( ax) sin(bx) cos(cx)dx
0

To evaluate the exponential sin-cos circular integral, we proceed in several
preliminary steps: (a) reduce the circular product term sin Acos B to unitary terms, (b)
convert to complex exponential form using real parts only as described above and combine

with real exponential & (c) apply gamma function formula  x m 1e  ax dx :
0


m 1
x          exp(ax) sin(bx) cos(cx)dx
0

 sin(b  c) x  sin(b  c) x 
  x m 1 exp( ax)                               dx                    (a)
              2              
0
               ie  i (b  c ) x  ie  i (b  c ) x 
 Re  x m 1 exp( ax)                                         dx       (b)

                   2                   

0
                           
1  m 1 a  (b  c )i x                                       
     Re   x ie                dx   x m 1ie a  (b  c )i x dx     (" )
2                                                                
0                           0                                
m            1                   1         
         Re i                 i                                        (c)
2       a  (b  c)i      a  (b  c)i m 
m
                                      

NOTE: The preliminary steps carry the bulk of the work.

Next (d) we find the complex conjugates of the denominators and multiply numerator
and denominator by that conjugate, and (e) use Euler’s Formula and De Moivre’s Theorem to
simplify. See Figure 1 which provides the basic information needed to solve this integral. A
diagram is always helpful to work from.

Page 7 of 19

r

(b  c)  r sin 



a = r cos 

Figure 1. Polar form of complex number a  b  c i
on a right triangle.

a  b  ci  re i  r cos  i sin 
a  b  cim  r me im  r m cos m  i sin m 
a2  b  c2  r 2
1

a 2  b  c 2    2
 r (positive square- root)
bc
tan  
a
        1     
b  c    tan ( ) 
  tan1
        2                2
 a   1                 1
tan ( )  tan ( )

In the derivation we must distinguish between the cases of b  c & b  c. It should be
clear in context. Continuing the reduction process….

Page 8 of 19

m            1                   1         
Re i                 i                  
2       a  (b  c)i      a  (b  c)i m 
m
                                      


m  
Re i
a  (b  c)i m         i
a  (b  c)i m          

2       a  (b  c)i  a  (b  c)i      a  (b  c)i m a  (b  c)i m 
m               m
                                                                       


m   a  (b  c)i m
Re i                   i
a  (b  c)i m 

2       2

 a  (b  c)
2 m
     2

a  (b  c)   2 m
     
m   r m e  im1    r m e  im 2 
      Re i           i               
2      
   r 2m            r 2m     
m   r m cos m1  i sin m1     r m cos m 2  i sin m 2  
        Re i                         i                              
2      
         r 2m                           r 2m              

m   sin m1 sin m 2 
                  
2  rm
            rm 


                                    
m      sin m1           sin m 2      
                                            Rem  0
m
2 
                                      
m
2         2                       
 a  (b  c) 2    a 2  (b  c) 2 2 

bc                             bc
where, 1  tan 1                     2  tan 1
a                               a

NOTE: This is the solution for the sin-cos case. It is involved but real parts complex
exponential analysis and the gamma function streamline a solution which would be
challenging to evaluate by standard real variable analysis. It is also easier than solving by the
classical Euler Formula definition of sine & cosine:

                                                                 eibx  e ibx  eicx  e icx 
m 1                                          m 1
x          exp( ax) sin(bx) cos(cx)dx   x             exp(ax)                              dx .
      2i              2       
0                                              0                                                

Page 9 of 19

Figure 2 shows a plot for this type of function which shows interesting features analogous to
damped oscillation. Convergence is evident by visual inspection3.

Figure 2. Example of a plot for function x 2 exp( x) sin( x) cos( x)
Source: http://www.wolframalpha.com

The other circular integrals for the unitary power case are not derived. But the same
process is used to evaluate them. We report the results of the calculations along with
exemplary plots.


m 1
x          exp(ax) cos(bx) cos(cx)dx
0
       1 b  c           1 b  c  
m   cos m tan         cos m tan         
a 
                      a 
                     Rem  0
2 
                                       
m                     m 
 a 2  b  c 2 2

a 2  b  c 2 2 


3
See Sokolnikoff (1939) for issues and conditions of convergence of improper integrals. In general integrals
will converge subject to the stated constraints.

Page 10 of 19

Figure 3. Example of a plot for function x 2 exp(2 x) cos( x) cos(2 x)
Source: http://www.wolframalpha.com


m 1
x          exp(ax) sin(bx) sin(cx)dx
0
       1 b  c           1 b  c  
m   cos m tan         cos m tan         
a 
                      a 
                             Rem  0
2 
                                   
m                     m 
 a 2  b  c 2 2

a 2  b  c 2 2 


Figure 4. Example of a plot for function x 2 exp(2 x) sin( x) sin(2 x)
Source: http://www.wolframalpha.com

Page 11 of 19

These integrals require more work to evaluate since more terms must be expanded,
combined and reduced. But real parts complex exponential analysis can provide a solution.

We will derive in detail the first second power integral—the sin 2 (bx) cos 2 (cx) case.

m 1
Derivation of     x          exp( ax) sin 2 (bx) cos 2 (cx)dx
0

The preliminary steps must be completed first: (a) reduce the quadratic power
circular terms and the circular product term to unitary terms, (b) convert to complex
exponential form using real parts only as described above and combine with real exponential

& (c) apply reduction formula  x m 1e  ax dx . This task is straightforward but tedious and
0
subject to algebraic errors. See next page.

Page 12 of 19


m 1
x           exp(ax) sin 2 (bx) cos 2 (cx)dx
0

 1  cos2bx   1  cos2cx  
  x m 1 exp( ax)                              dx
       2              2       
0

1  cos(2cx)  cos(2bx)  cos(2bx) cos(2cx) 
  x m 1 exp( ax)                                              dx
                     4                      
0

1                                             cos(2b  2c) x  cos(2b  2c) x 
  x m 1 exp( ax)1  cos(2cx)  cos(2bx)                                   dx
4                                                            2                
0
                                                                  
1                       1                                1
  x m 1 exp( ax) dx   x m 1 exp( ax) cos(2cx) dx   x m 1 exp( ax) cos(2bx) dx
4                       4                                4
0                             0                                      0
                                            
1 m 1                             1 m 1

8  x exp(ax) cos(2b  2c) x dx  8  x exp(ax) cos(2b  2c) x dx
0                                             0
                                                       
1                         1            a  2ci x     1            a  2bi x
 Re  x m 1 exp(ax) dx   x m 1e              dx   x m 1e              dx
4                         4                            4
0                            0                          0
                                  
1 m 1 a  2b  2c i x     1            a  2b  2c i x
     x e                      dx   x m 1e                      dx
8                               8
0                                  0
m   1           1           1      1        1            1        1           
        Re                                                                      
4      a

m
a  2bi m a  2cim 2 a  2b  2c i m 2 a  2b  2c i m 


Having completed the tedious construction of preliminary steps, we now find the ratio
of complex conjugates as before, and go to the polar diagrams. See Figure 5. There will be
one diagram for each for the complex numbers, a  2bi, a  2ci, a  (2b  2c)i, a  (2b  2c)i .
In the derivation we must distinguish between each complex number. It should be clear in
context.

Page 13 of 19

2        2
Figure 5. Polar form of complex numbers for sin (bx) cos (cx) integral.

r                                                      r

2b  r sin                                            2c  r sin 

                                                      

a = r cos                                             a = r cos 

a  2bim  r meim                                 a  2cim  r meim
a 2  4b2  r 2                                       a 2  4c 2  r 2

r                                                      r

( 2b  2c)  r sin                                    (2b  2c)  r sin 

                                                      

a = r cos                                             a = r cos 

a  b  cim  r meim                              a  b  cim  r meim
a 2  2b  2c2  r 2                                  a 2  2b  2c2  r 2

NOTE: The values of r &  for each case must be distinguished in the derivation steps.
Refer to Fig. 1 for other definitions.

Continuing the reduction process….

Page 14 of 19

 1           a  2bi m                  a  2cim                                 
 m                                                                                
m   a       a  2bi m a  2bi m a  2cim a  2cim                                 
Re                                                                                      
4      1           a  2b  2c i m            
1       a  2b  2c i m            
 2 a  2b  2c i m a  2b  2c i m 2 a  2b  2c i m a  2b  2c i m 
                                                                                     
                                                                                        
m   1       r m e  im1     r m e  im 2   1      r m e  im 3        1   r m e  im 4    
       Re                                                                
4       am

2

a  4b    2 m
   2
a  4c    2 m 2 2
       
a  2b  2c           
2 m 2 2

a  2b  2c   2 m
 
m   1 r m cos m1  i sin 1  r m cos m 2  i sin  2  1 r m cos m 3  i sin  3  1 r m cos m 4  i sin  4  
       Re                                                                                                               
4       am
                r 2m                       r 2m                2        r 2m              2     r 2m              

m   1 cos m1 cos m 2 1 cos m 3 1 cos m 4 
                                               
4 am        rm           rm      2 rm           2 rm         
                                                                         
m   1   cos m1      cos m 2     1    cos m 3         1    cos m 4       
        m                                              
m
4 a
            
2
            2
                      
m             m                     m
2 2 

2
a  4b 2 2    2
a  4c 2 2
a  2b  2c 
2            2 2
a  2b  2c 
2


where,
2b               2c              2b  2c                2b  2c
1  tan  1       ,  2  tan 1 ,  3  tan 1         ,  4  tan 1
a                a                  a                      a


Thus,  x m 1 exp(ax) sin 2 (bx) cos 2 (cx)dx
0
                  2b              2c                   2b  2c                   2b  2c 
      cos m tan 1  cos m tan 1          cos m tan 1              cos m tan 1         
m   1                 a               a  1                   a  1                       a 
                                                                          
                                                                   
4 am                    m                 m
2                    m      2                     m



2
a  4b   2 2

a 2  4c 2
2
a  2b  2c 
2
           2 2
a 2  2b  2c 22

Rea, m  0

NOTE: Particular solutions can be obtained from the general solution; e.g., set
m  1, or c  0, but not b  0, or combination of m  1 & c  0.

________________

Page 15 of 19

Figure 6 shows a plot for this family of functions. It is similar to the first degree case (Fig. 2)
except in the quadratic case all amplitudes are positive.

Figure 6. Example of a plot for function x 2 exp(2 x) sin 2 (2 x) cos 2 (2 x)
Source: http://www.wolframalpha.com

_____

The other circular integrals for the quadratic power case are not derived. But the same
process is used to evaluate them. We report the results of the calculations and show
exemplary plots.


m 1
x          exp(ax) cos 2 (bx) cos 2 (cx)dx
0
                 2b            2c                   2b  2c                   2b  2c 
 1   cos m tan 1  cos m tan 1        cos m tan 1              cos m tan 1         
m                    a             a  1                   a  1                      a 
                                                                        
4 am
                        2
                   2
                    
m               m                           m                           m

2
a  4b   2 2       2
a  4c 2 2
a  2b  2c 
2             2 2
a  2b  2c 
2             2 2

Rea, m  0

Page 16 of 19

Figure 7. Example of a plot for function x 2 exp(2 x) cos 2 ( x) cos 2 (2 x)
Source: http://www.wolframalpha.com


m 1
x          exp(ax) sin 2 (bx) sin 2 (cx)dx
0
                 2b            2c                   2b  2c                   2b  2c 
 1   cos m tan 1  cos m tan 1        cos m tan 1              cos m tan 1         
m                    a             a  1                   a  1                      a 
                                                                       
4 am
                        2
                    2
                    
m               m                           m                           m

2
a  4b   2 2       2
a  4c 2 2
a  2b  2c 
2             2 2
a  2b  2c 
2             2 2

Rea, m  0

Figure 8. Example of a plot for function x 2 exp(2 x) sin 2 ( x) sin 2 (2 x)
Source: http://www.wolframalpha.com

Page 17 of 19

Inverse Models

We call an “inverse” model integrals of the form,

              cos n bx  cos p cx 

1
exp( ax)                          dx
 xm            sin n bx   sin p cx  
0             
                           


This model is evaluated by setting x m 1  x  m in the class of integrals,

                  cos n bx  cos p cx 

x m 1 exp(ax)                          dx
                  sin n bx   sin p cx  
0                
                           


and using Euler’s Reflection Formula,


m 1  m                  m  0,1,2,...
sin m

For example, the first degree integral,

                                         
1
 x m exp(ax) sin bx  coscx dx   x          exp(ax) sin bx  coscx dx
m

0                                         0
                                         
1              sin(1  m)1           sin(1  m) 2 
                                                                       Rem  0, not integer 
2 m  sin m               1 m                  1 m 
2
        2
 a  (b  c) 2              
a 2  (b  c) 2 2 


bc             bc
where, 1  arctan     2  arctan   
 a               a 

A plot of the function for specified parameters is given in Fig. 9.

Page 18 of 19

1
Fig. 9. Example of a plot of            exp(2 x) sin 2 x  cos(2 x)
x
Source: http://www.wolframalpha.com

________

Integrals for Higher Powers

Powers greater than quadratic require general case identities for odd and even
integers. These are summarized in the paper “Algebraic-exponential-trigonometric integrals
of integer powers,” Appendix I. In general, there appears to be about 3d  1 integral
components where d is the degree of the circular terms; if n  p  4, we expect about 11
integral components in the expansion. A computer program could be created for the odd-
even expansion formulas in the referenced Appendix, the circular product relations and the
real only complex exponential for cosine, sine.

Page 19 of 19

References

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

Carr, G.S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea
Publishing Co., 1970.

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.

Gradshteyn, I.S. and I.M. Ryzhik. Table of Integrals, Series, and Products (7th Edition). Alan
Jeffrey and Daniel Zwillinger, Editors. NY: Academic Press, 2007.
Errata: http://www.mathtable.com/errata/gr7_errata.pdf

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

O’Brien, F. J., Jr. “  x m e   x dx and related integrals, 2nd ed.”, Jan. 3, 2013,
n

http://www.docstoc.com/profile/waabu.

_______.“General Cosine and Sine Integral of Powers,” Apr. 22, 2013,
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_______. “Algebraic–Exponential–Trigonometric Integrals of Integer Powers, ” May 15,
2013. http://www.docstoc.com/profile/waabu.

Sokolnikoff, I. S. Advanced Calculus. New York: McGraw Hill Book Co., 1939.

Spiegel, M. R. Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill,
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