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Bessel Functions of Fractional Order in .NET Creator barcode pdf417 in .NET Bessel Functions of Fractional Order




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6.7 Bessel Functions of Fractional Order use .net pdf417 generator toincoporate pdf 417 for .net BIRT Reporting Tools sum=ff; e=exp(e); p= .net framework pdf417 2d barcode 0.5*e/gampl; p0 .

q=0.5/(e*gammi); q0 . c=1.

0; d=x2*x2; sum1=p; for (i=1;i<=MAXIT;i++) { ff=(i*ff+p+q)/(i*i-xmu2); c *= (d/i); p /= (i-xmu); q /= (i+xmu); del=c*ff; sum += del; del1=c*(p-i*ff); sum1 += del1; if (fabs(del) < fabs(sum)*EPS) break; } if (i > MAXIT) nrerror("bessk series failed to converge"); rkmu=sum; rk1=sum1*xi2; } else { Evaluate CF2 by Steed s algorithm b=2.0*(1.0+x); ( 5.

2), which is OK because there d=1.0/b; can be no zero denominators. h=delh=d; q1=0.

0; Initializations for recurrence (6.7.35).

q2=1.0; a1=0.25-xmu2; q=c=a1; First term in equation (6.

7.34). a = -a1; s=1.

0+q*delh; for (i=2;i<=MAXIT;i++) { a -= 2*(i-1); c = -a*c/i; qnew=(q1-b*q2)/a; q1=q2; q2=qnew; q += c*qnew; b += 2.0; d=1.0/(b+a*d); delh=(b*d-1.

0)*delh; h += delh; dels=q*delh; s += dels; if (fabs(dels/s) < EPS) break; Need only test convergence of sum since CF2 itself converges more quickly. } if (i > MAXIT) nrerror("bessik: failure to converge in cf2"); h=a1*h; rkmu=sqrt(PI/(2.0*x))*exp(-x)/s; Omit the factor exp( x) to scale rk1=rkmu*(xmu+x+0.

5-h)*xi; all the returned functions by exp(x) } for x XMIN. rkmup=xmu*xi*rkmu-rk1; rimu=xi/(f*rkmu-rkmup); Get I from Wronskian. *ri=(rimu*ril1)/ril; Scale original I and I .

*rip=(rimu*rip1)/ril; for (i=1;i<=nl;i++) { Upward recurrence of K . rktemp=(xmu+i)*xi2*rk1+rkmu; rkmu=rk1; rk1=rktemp; } *rk=rkmu; *rkp=xnu*xi*rkmu-rk1; }. 6. . Special Functions Airy Functions For positive x, the visual .net barcode pdf417 Airy functions are de ned by Ai(x) = Bi(x) = where 1 x K1/3 (z) 3 x [I1/3 (z) + I 1/3 (z)] 3 z= (6.7.

41) (6.7.42).

2 3/2 x (6.7.43) 3 B y using the re ection formula (6.

7.40), we can convert (6.7.

42) into the computationally more useful form 2 1 Bi(x) = x I1/3 (z) + K1/3 (z) (6.7.44) 3.

so that Ai and Bi ca PDF-417 2d barcode for .NET n be evaluated with a single call to bessik. The derivatives should not be evaluated by simply differentiating the above expressions because of possible subtraction errors near x = 0.

Instead, use the equivalent expressions x Ai (x) = K2/3 (z) 3 (6.7.45) 2 1 Bi (x) = x I2/3 (z) + K2/3 (z) 3 The corresponding formulas for negative arguments are x 1 Ai( x) = J1/3 (z) Y1/3 (z) 2 3 x 1 J1/3 (z) + Y1/3 (z) Bi( x) = 2 3 x 1 Ai ( x) = J2/3 (z) + Y2/3 (z) 2 3 x 1 J2/3 (z) Y2/3 (z) Bi ( x) = 2 3.

#include <math.h& gt; #define PI 3.1415927 #define THIRD (1.

0/3.0) #define TWOTHR (2.0*THIRD) #define ONOVRT 0.

57735027 void airy(float x, float *ai, float *bi, float *aip, float *bip) Returns Airy functions Ai(x), Bi(x), and their derivatives Ai (x), Bi (x). { void bessik(float x, float xnu, float *ri, float *rk, float *rip, float *rkp); void bessjy(float x, float xnu, float *rj, float *ry, float *rjp, float *ryp); float absx,ri,rip,rj,rjp,rk,rkp,rootx,ry,ryp,z; absx=fabs(x); rootx=sqrt(absx); z=TWOTHR*absx*rootx; if (x > 0.0) { bessik(z,THIRD,&ri,&rk,&rip,&rkp); *ai=rootx*ONOVRT*rk/PI;.

(6.7.46).

6.7 Bessel Functions of Fractional Order *bi=rootx*(rk/PI+2.0 .net vs 2010 barcode pdf417 *ONOVRT*ri); bessik(z,TWOTHR,&ri,&rk,&rip,&rkp); *aip = -x*ONOVRT*rk/PI; *bip=x*(rk/PI+2.

0*ONOVRT*ri); } else if (x < 0.0) { bessjy(z,THIRD,&rj,&ry,&rjp,&ryp); *ai=0.5*rootx*(rj-ONOVRT*ry); *bi = -0.

5*rootx*(ry+ONOVRT*rj); bessjy(z,TWOTHR,&rj,&ry,&rjp,&ryp); *aip=0.5*absx*(ONOVRT*ry+rj); *bip=0.5*absx*(ONOVRT*rj-ry); } else { Case x = 0.

*ai=0.35502805; *bi=(*ai)/ONOVRT; *aip = -0.25881940; *bip = -(*aip)/ONOVRT; } }.

Spherical Bessel Functions For integer n, spher PDF 417 for .NET ical Bessel functions are de ned by jn (x) = yn (x) = Jn+(1/2) (x) 2x Yn+(1/2) (x) 2x. (6.7.47).

They can be evaluate PDF 417 for .NET d by a call to bessjy, and the derivatives can safely be found from the derivatives of equation (6.7.

47). Note that in the continued fraction CF2 in (6.7.

3) just the rst term survives for = 1/2. Thus one can make a very simple algorithm for spherical Bessel functions along the lines of bessjy by always recursing jn down to n = 0, setting p and q from the rst term in CF2, and then recursing yn up. No special series is required near x = 0.

However, bessjy is already so ef cient that we have not bothered to provide an independent routine for spherical Bessels.. #include <math.h& gt; #define RTPIO2 1.2533141 void sphbes(int n, float x, float *sj, float *sy, float *sjp, float *syp) Returns spherical Bessel functions jn (x), yn (x), and their derivatives jn (x), yn (x) for integer n.

{ void bessjy(float x, float xnu, float *rj, float *ry, float *rjp, float *ryp); void nrerror(char error_text[]); float factor,order,rj,rjp,ry,ryp; if (n < 0 . x <= 0.0) nrerro .net vs 2010 barcode pdf417 r("bad arguments in sphbes"); order=n+0.

5; bessjy(x,order,&rj,&ry,&rjp,&ryp); factor=RTPIO2/sqrt(x); *sj=factor*rj; *sy=factor*ry; *sjp=factor*rjp-(*sj)/(2.0*x); *syp=factor*ryp-(*sy)/(2.0*x); }.

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