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[VA] SRC #008 - 2021 is here !
01-06-2021, 01:20 PM
Post: #24
RE: [VA] SRC #008 - 2021 is here !
(01-04-2021 04:16 PM)Albert Chan Wrote:  Post #10, we transform I by using substitution: y = pi/2-x, dy = -dx

We can continue substitutions: t = tan(y), dt = (1+t^2) dy
This shift integration limit from y = 0 .. pi/2, to t = 0 .. infinity

One more substitution, t = (1-u)/u, dt = (-1/u^2) du
This shift back integration limit back to finite: u = 0 .. 1

Another way is made use of symmetry.
Code:

I = ∫ f(y) dy, y = -pi/2 .. pi/2
  = ∫ [f(y) + f(-y)] dy, y = 0 .. pi/2         // Fold
  = ∫ g(y) dy, y = 0 .. pi/2                         
  = ∫ [g(y) + g(pi/2-y)] dy, y = 0 .. pi/4     // Fold again

Now, let t = tan(y), dt = (1+t^2) dy
This shift integration limit from y = 0 .. pi/4, to t = 0 .. 1

Integrand looks more complicated, but it actually converge faster !

Note: t = tan(y), tan(pi/2-y) = cot(y) = 1/t

10 INPUT "K? ";K @ C=2^(-1/K) @ P=.0000000001
20 DEF FNF(T,F)=LN((.5+(T+C)^K)*(.5+(F+C)^K)/((.5+ABS(T-C)^K)*(.5+(F-C)^K)))/T
30 T=TIME
40 S1=INTEGRAL(0,C,P,FNF(IVAR,1/IVAR))/K
50 S2=INTEGRAL(C,1,P,FNF(IVAR,1/IVAR))/K
60 DISP S1;"+";S2;"=";S1+S2,TIME-T

>RUN
k? 2.021
 1.88084223982 + .586558860435 = 2.46740110026                  .44
>RUN
k? 1
 1.47906894032 + .98833215996 = 2.46740110028                   .44
>RUN
k? 2
 1.87491615102 + .59248494925 = 2.46740110027                   .44
>RUN
k? 3
 2.08306009111 + .38434100916 = 2.46740110027                   .44
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Messages In This Thread
RE: [VA] SRC #008 - 2021 is here ! - Gene - 01-02-2021, 01:49 AM
RE: [VA] SRC #008 - 2021 is here ! - robve - 01-03-2021, 06:33 PM
RE: [VA] SRC #008 - 2021 is here ! - robve - 01-05-2021, 03:39 AM
RE: [VA] SRC #008 - 2021 is here ! - Albert Chan - 01-06-2021 01:20 PM
RE: [VA] SRC #008 - 2021 is here ! - Gene - 01-04-2021, 05:56 PM
RE: [VA] SRC #008 - 2021 is here ! - Gene - 01-04-2021, 07:24 PM
RE: [VA] SRC #008 - 2021 is here ! - Gene - 01-06-2021, 02:54 PM
RE: [VA] SRC #008 - 2021 is here ! - EdS2 - 01-08-2021, 01:32 PM



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