Infinite range variable change

  

   Note that 0<t<1

So that

Arctan g(x)=1/(1+x2)

Let

  Compare which had 1/(1+x)2

So that

Solve for r to find

So that

Midpoint trap rule evaluation of the integral infinite range

Interesting integral

Dwight 856.31[i]

 

for\midpt.for

Using N points

      IMPLICIT REAL*8(A-H,O-Z)

      PARAMETER (N=5000)

      COMMON/PASS/A,B

      DATA PI/3.141592653589793D0/

C *** MID-POINT

      A=17

      B=23

      AINT=0

      H=1D0/N

      DO I=1,N

        T=H*(I-.5D0)

        R=TAN(PI*(T-.5D0))

        F=(1+R**2)*FTEST(R)

        AINT=AINT+F

      ENDDO

      AINT=AINT*H*PI

      ANAL=PI/(A*B*(A+B))

      PRINT*,' NUMERICAL FULL INTEGRAL ',AINT

      PRINT*,' ANALYTIC FULL INTEGRAL  ',ANAL

      READ(*,'(A)')

      END

      FUNCTION FTEST(X)

      IMPLICIT REAL*8(A-H,O-Z)

      COMMON/PASS/A,B

      FTEST=1/((A*A+X*X)*(B*B+X*X))

      RETURN

      END

 

FULL INTEGRAL  5000 pts  2.008690954980680D-004  mid point

2.008690954980690D-004  analytical

compare this accuracy with #Semi_result

The difference is due to the fact that the integrals at the end of the region are smooth.  The function can be periodically continued. – more on this later.

 

Figure 2 Full integration range.

Figure 3  Lower interval

Figure 4  Upper interval

Assignment – infinite integrals

  1. Evaluate for other A and B’s
  2. Evaluate Dwight 856.21
  3. Evaluate  Dwight 858.652
  4. Evaluate Dwight 861.20

 

Note that c. does not go to zero as t à 0 and 1.  In addition its accuracy is somewhat limited owing to an infinite set of oscillations.

The 1/r2 transform

The function g(r)=1/(a2+r2) has an integral

So that

Solve for r(t) to find

      

So that

FOR\midptal.for

 


[i] H.B.Dwight, Tables of Integrals and other Mathematical Data, Macmillan 1961