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General Cosine and Sine Integral of Powers

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               General Cosine and Sine Integral of Powers
                                                      ╬

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

 Francis J. O’Brien, Jr.                                                                  April 22, 2013

Introduction
        In this paper we show the derivation for two general algebraic-trigonometric integrals
using complex variables and the gamma function1. The new integral expressed in combined
form is:

                                             
                         
                   m cos x 
                              n          cos 
                                        2       m 1
                 x sin x n dx  n      ,   n ,  ,   0, n  0,
                           
                0                       sin   
                                            2 
                                                  
where      is the gamma function. See L. Bers, Calculus (pp. 402-403) for elementary
properties.

Complex variables often simplify trigonometric integral evaluation2.


    Proof Outline

    Familiar elementary complex variable relations and operations:

       Euler’s Formula
         eix  cos x  i sin x
         e ix  cos x  i sin x

1
  Submitted to 8th edition of Gradshyetn and Rhyzik (GR), Table of Integrals, Series and Products for Section
3.761, “Trigonometric Functions of More Complicated Arguments Combined with Powers”. NOTE: One
website, http://quickmath.com, was unable to do the definite form or the indefinite form (a
complicated indefinite form solution was given at the Mathematica website,
http://integrals.wolfram.com/index.jsp).
2
  G. S. Carr, Formulas and Theorems in Pure Mathematics, solves similar integrals using complex
variables and gamma function; see Theorems 2577–2579. Theorem 2577 is derived below to
demonstrate complexification as a means to solve an integral computers cannot evaluate.

 Francis J. O’Brien, Jr., 2013 <> Aquidneck Indian Council<> All rights reserved.
                                                                                                Page 2 of 9



       By definition from Euler’s Formula,

                                                          eix  e ix
                                                cos x                .
                                                               2

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

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


                          sin x  Re
                                      i
                                        
                                      e ix
                                          Re  ie ix or Re  e
                                        
                                                            
                                                               ix
                                                                  i
                                                                               
                                                                               
                                                                                       
                                                                                 Re ie  ix
                                                                            
                                                e  ix 
                               Use sin x  Re 
                                               
                                                          Re ie  ix
                                                      i 
                                                                            for convergence of
                                                       
                                integrals of negative exponentials


NOTE: the simplified real only formulas are valid for sine & cosine powers of 1. For integer
                                                      1  cos(2 x)
powers higher than 1 use identities such as sin 2 x               , then power of 1 real
                                                           2
relations. A future paper will demonstrate the technique.

       Euler’s identity, ei  1
                  i
             o   e2      11 2  i
                  i
                                        
             o   e 2  i  cos   i sin 
                                    2            2 
                   i
                                        
             o   e 2  i  cos   i sin 
                                         2             2 

 Francis J. O’Brien, Jr., 2013 <> Aquidneck Indian Council<> All rights reserved.
                                                                                                     Page 3 of 9


       Complex conjugation

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

              o           1.
                   i  i   i 2


       Complexifying3 the integral by real parts only analysis


    The following algebraic-exponential formula from Gradshteyn & Rhyzik is used to
    simplify the complexified integrals:

         
                                              m 1
         x
              m  x n
               e         dx              ,          ,  ,   0, n  0   Formula 3.326.2, GR (7th ed.)4
                                      
         0                       n                n




    Derivation

    Inserting the above relations and operations, and simplifying, the complexified cosine
integral is:


                                        dx  Re  x meix dx  Re n i   ii 
                                                                                      
                                                                     n
                          x cos x
                             m                n

                         0                                0                         
                                     i           
                                 2                    
                                 Re e                   cos ,
                             n                    n       2 
                                                   
                                 m 1
                                    , Re ,   0, n  0
                                  n

or, calculate sum of integrals by classical definition of cosine,




3
  A good YouTube lecture from MIT is at “18.03 Differential Equations” (Lecture 6).
                m  x n
               
4
  Derived in “ x e       dx and related integrals, 2nd ed.”, Jan. 3, 2013.
http://www.docstoc.com/profile/waabu

 Francis J. O’Brien, Jr., 2013 <> Aquidneck Indian Council<> All rights reserved.
                                                                                                              Page 4 of 9



                                                         
                                       
                                       x       cos x n dx
                                            m

                                       0
                                                    i x n
                                                        m e
                                                                         n
                                                                 e  i x     
                                        Re  x                            dx
                                                              2            
                                            0                              
                                         1          1 i             1  i              
                                       
                                         2 n  
                                                 
                                                 
                                                   Re
                                                        i i       
                                                                    Re
                                                                           i  i            
                                                 
                                              cos .
                                         n        2 

                                   ____________

The complexified sine integral is similar. Not all the steps are specified:


                                                                                
                                                         
                                                         x            sin x n dx
                                                               m

                                                         0
                                                                           
                                                          Re i  x m e  ix dx
                                                                                       n


                                                                           0
                                                                  
                                                               sin  
                                                                       
                                                                     2 
                                                             n

or, calculate difference of integrals by classical definition of sine,

                                                   
                                        i x n  e  i x n                     
                              Re  x    m e                                         dx    sin   
                                                                                                      
                                 0
                                     
                                     
                                                 2i                                 
                                                                                          n        2 




               cosx        dx
             
                        2
Example:
              0

        In the general solution,


                                                                        
                                                 
                                                                                             
                                                    x       cos x n dx 
                                                         m
                                                                                            cos ,
                                                    0                                  n      2 



 Francis J. O’Brien, Jr., 2013 <> Aquidneck Indian Council<> All rights reserved.
                                                                                               Page 5 of 9


                                           2   1
set m  0,   1, n  2,   1 / 2, cos          to give the solution by the “short cut” or
                                       4   2     2
standard method:


                                  cosx        dx
                                 
                                           2

                                 0
                                                  
                                           ix 2
                                        e      e               ix 2 
                                                                       
                                                                                 
                                                                         dx  Re  e  ix dx
                                                                                         2
                                  Re                                
                                      0
                                             2                        
                                                                                0

                                    1
                                    
                                            
                                    cos 
                                      2
                                     2     4
                                      2
                                 
                                      4
                                     1 
                                         0.62665706865,
                                     2 2

       1
where     .
       2

Example: The cosine Fresnel integral is defined to be:

                                                          z
                                                                 2 
                                                           cos 2 t
                                                               
                                                                     dt.
                                                                     
                                                          0



It cannot be solved in closed form unless z   in which case the general integral

             

                             
x       cos x n dx 
     m
                            cos  holds (for                              .
                               2 
0                        n
                                                        2   1
           Set m  0,       , n  2,   1 / 2, cos          :
                            2                        4   2     2

                                                      
                                                               2      1
                                                       cos 2 t
                                                           
                                                                    dt  .
                                                                        2
                                                      0

The sine Fresnel integral

 Francis J. O’Brien, Jr., 2013 <> Aquidneck Indian Council<> All rights reserved.
                                                                                                               Page 6 of 9

                                                   z
                                                            2 
                                             lim
                                            z 
                                                  sin 2 t
                                                      
                                                                dt
                                                                
                                                   0

gives the same solution.

                                                                                         
NOTE: Theorem 2577 (p. 384) in G.S Carr,               x
                                                            n 1  ax
                                                               e        cosbx dx and    x
                                                                                                  n 1  ax
                                                                                                       e       sin bx dx .
                                                       0                                  0
Online integrators cannot solve these definite integrals; e.g., http://www.quickmath.com/

       These integrals are useful because they use clever manipulations of complex variables
and serve as a general form from which specific cases can be derived.

        Figure 1 provides the essential information needed for complex variables to evaluate
the integrals.




                                                                                                   i
                                                                                 a  bi  re

                                                                                                        in
                                r                                                a  bi n
                                                                                              r e
                                                                                                   n



                                             b = r sin                           2
                                                                                 a b  r
                                                                                          2       2



                                                                                             b
                                                                                 tan  
                                                                                              a
                         a = r cos 
                                                                                               1 b 
                                                                                    tan         a
          Figure 1. Polar form of complex number a  bi                                            
          on a right triangle. The form a  bi would
          lie in Quadrant IV.


NOTE: it is more traditional to use x & y vs. a & b for polar coordinates.

                                                                                      
Re-expressing in complex variable form and applying Form. 3.326.2,  x m e  x dx, the cosine
                                                                                                         n


                                                                                      0
integral is:




 Francis J. O’Brien, Jr., 2013 <> Aquidneck Indian Council<> All rights reserved.
                                                                                                                                    Page 7 of 9

                                                                                      
                                                                                             n 1  a  bi x              n 
             x
                  n 1  ax
                      e       cosbx dx         x
                                                           n 1  ax  ibx
                                                              e      e           dx    x       e               dx 
             0                                    0                                     0                               a  bi n

NOTE: With complex variables we have reduced a complicated algebraic-exponential-
trigonometric function to an algebraic-exponential function which is more manageable.

   The solution must be given in terms of the real number parameters a, b & n. Thus,
conjugating, inserting the auxiliary polar equations, and simplifying provides a string of
reductions:


                                                                   
                                              n 1  a  bi x
                       x e cosbxdx  Re  x e                dx  n  Re
                         n 1  ax                                                                                      1
                      0                                              0                                            a  bi n
                               n  Re
                                                   1     a  bi n               n  Re
                                                                                                    a  bi n
                                              a  bi n a  bi n                            a  bi a  bi n
                                                        n  Re
                                                                                 1
                                                                                      rei n
                                                                         a 2  b2 n
                                   n  Re
                                                      rn
                                                       2n
                                                          cos n  i sin n   n cosnn
                                                   r                                                    r
                                              tan   b a
                                                                   n                       b 
                                                                                cos n tan 1 .
                                                                                             a 
                                                                     n
                                                               2   2 2                 
                                                              a b

        
                                                 n                           b 
Thus,   x
             n 1  ax
                  e       cosbx dx                             cos n  tan 1 .
                                              a 2  b2                         a 
                                                              n
        0                                                     2
                                                                        



                                                                                                                              
                              
NOTE: Solution for            x
                                   n 1  ax
                                      e          sin bx dx is similar, defining sin(bx )  Re ie  ibx .
                              0
NOTE: If b  0, Theor. 2577 gives

                                          
                                                                         n                   n 
                                          x
                                                n 1  ax
                                                       e      dx                cos n0  
                                                                             n
                                          0                              a                        an
since arctan(0)  0, cos(0)  1.



 Francis J. O’Brien, Jr., 2013 <> Aquidneck Indian Council<> All rights reserved.
                                                                                                    Page 8 of 9


NOTE: If a  0, Theor. 2577 gives

                                          
                                          x
                                               n 1
                                                       cosbx dx
                                          0
                                                  
                                           Re  x n 1e  bi x dx
                                                   0
                                              n                    b 
                                                 cos n  lim tan 1  
                                             bn         a  0         a 

                                          
                                            n 
                                             b n
                                                            
                                                  cos n tan 1   ,       
                                                                                
and recalling the limit of arctangent at infinity, lim tan 1 x                     1.57 , then
                                                                x             2

                                      
                                                                    n   
                                      x
                                           n 1
                                                  cosbx dx          cos n 
                                      0                             bn     2




                                          Figure 2. Plot of y  tan 1 x .



NOTE: If n  1, the first step in the derivation of Theor. 2577 gives an immediate answer:

                   
                                                     1             ( a  bi )
                               cosbx dx  Re
                         ax                                                            a
                   e                              a  bi 
                                                              Re                   
                                                                  a  bi a  bi  a  b 2
                                                                                      2
                   0
 Francis J. O’Brien, Jr., 2013 <> Aquidneck Indian Council<> All rights reserved.
                                                                                     Page 9 of 9


where  1 =1.

Clearly, Theorem 2577 is a useful integral. Why can’t computers solve it?

        A future paper will demonstrate the solution to integrals of the form

                                                     cos n  x 
                                                                   
                                         x m 1e x              dx
                                        0             sin n  x  
                                                                   

The approach is modeled on the solution of Theorem 2577.




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

				
DOCUMENT INFO
Description: In this paper we show the derivation for two general algebraic-trigonometric integrals using elementary complex variables and the gamma function. Typos corrected in previous two uploads.