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Fresnel Integrals, Cosine and Sine Integrals in Software Generate pdf417 2d barcode in Software Fresnel Integrals, Cosine and Sine Integrals




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6.8 Fresnel Integrals, Cosine and Sine Integrals generate, create pdf417 none with software projects bar code 128 Thus the precis PDF 417 for None ion of the routine frenel may be limited by the precision of the library routines for sine and cosine for large x.. Complex frenel( const Doub x) { Computes the Fresnel integrals S.x/ and C.x/ for all real x.

C.x/ is returned as the real part of cs and S.x/ as the imaginary part.

static const Int MAXIT=100; static const Doub PI=3.141592653589793238, PIBY2=(PI/2.0), XMIN=1.

5, EPS=numeric_limits<Doub>::epsilon(), FPMIN=numeric_limits<Doub>::min(), BIG=numeric_limits<Doub>::max()*EPS; Here MAXIT is the maximum number of iterations allowed; EPS is the relative error; FPMIN is a number near the smallest representable oating-point number; BIG is a number near the machine over ow limit; and XMIN is the dividing line between using the series and continued fraction. Bool odd; Int k,n; Doub a,ax,fact,pix2,sign,sum,sumc,sums,term,test; Complex b,cc,d,h,del,cs; if ((ax=abs(x)) < sqrt(FPMIN)) { Special case: Avoid failure of convergence cs=ax; test because of under ow. } else if (ax <= XMIN) { Evaluate both series simultaneously.

sum=sums=0.0; sumc=ax; sign=1.0; fact=PIBY2*ax*ax; odd=true; term=ax; n=3; for (k=1;k<=MAXIT;k++) { term *= fact/k; sum += sign*term/n; test=abs(sum)*EPS; if (odd) { sign = -sign; sums=sum; sum=sumc; } else { sumc=sum; sum=sums; } if (term < test) break; odd=!odd; n += 2; } if (k > MAXIT) throw("series failed in frenel"); cs=Complex(sumc,sums); } else { Evaluate continued fraction by modi ed pix2=PI*ax*ax; Lentz s method ( 5.

2). b=Complex(1.0,-pix2); cc=BIG; d=h=1.

0/b; n = -1; for (k=2;k<=MAXIT;k++) { n += 2; a = -n*(n+1); b += 4.0; d=1.0/(a*d+b); Denominators cannot be zero.

cc=b+a/cc; del=cc*d; h *= del; if (abs(real(del)-1.0)+abs(imag(del)) <= EPS) break; } frenel.h.

6. Special Functions /2 1.5 Si(x) 1. cosine and sine integrals Ci(x). Figure 6.8.2.

S barcode pdf417 for None ine and cosine integrals Si.x/ and Ci.x/.

. if (k > MAXI T) throw("cf failed in frenel"); h *= Complex(ax,-ax); cs=Complex(0.5,0.5) *(1.

0-Complex(cos(0.5*pix2),sin(0.5*pix2))*h); } if (x < 0.

0) cs = -cs; return cs; } Use antisymmetry.. 6.8.2 Cosine and Sine Integrals The cosine and sine integrals are de ned by Z Ci.x/ D Z Si.x/ D C ln x C cos t t dt (6.8.6).

sin t dt t and are plotted Software pdf417 in Figure 6.8.2.

Here 0:5772 : : : is Euler s constant. We only need a way to calculate the functions for x > 0, because Si. x/ D Si.

x/; Ci. x/ D Ci.x/ i (6.

8.7). Once again we can evaluate these functions by a judicious combination of power 6.8 Fresnel Integrals, Cosine and Sine Integrals series and comp lex continued fraction. The series are Si.x/ D x Ci.

x/ D x5 x3 C 3 3 5 5 x2 x4 C ln x C C 2 2 4 4 . (6.8.8).

The continued f raction for the exponential integral E1 .ix/ is E1 .ix/ D Ci.

x/ C i Si.x/ =2 1 1 1 2 2 D e ix ix C 1 C ix C 1 C ix C 1 12 22 D e ix 1 C ix 3 C ix 5 C ix (6.8.

9). The even form Software PDF-417 2d barcode of the continued fraction is given in the last line and converges twice as fast for about the same amount of computation. A good crossover point from the alternating series to the continued fraction is x D 2 in this case. As for the Fresnel integrals, for large x the precision may be limited by the precision of the sine and cosine routines.

. Complex cisi(co nst Doub x) { Computes the cosine and sine integrals Ci.x/ and Si.x/.

The function Ci.x/ is returned as the real part of cs, and Si.x/ as the imaginary part.

Ci.0/ is returned as a large negative number and no error message is generated. For x < 0 the routine returns Ci.

x/ and you must supply the i yourself. static const Int MAXIT=100; Maximum number of iterations allowed. static const Doub EULER=0.

577215664901533, PIBY2=1.570796326794897, TMIN=2.0, EPS=numeric_limits<Doub>::epsilon(), FPMIN=numeric_limits<Doub>::min()*4.

0, BIG=numeric_limits<Doub>::max()*EPS; Here EULER is Euler s constant ; PIBY2 is =2; TMIN is the dividing line between using the series and continued fraction; EPS is the relative error, or absolute error near a zero of Ci.x/; FPMIN is a number close to the smallest representable oating-point number; and BIG is a number near the machine over ow limit. Int i,k; Bool odd; Doub a,err,fact,sign,sum,sumc,sums,t,term; Complex h,b,c,d,del,cs; if ((t=abs(x)) == 0.

0) return -BIG; Special case. if (t > TMIN) { Evaluate continued fraction by modi ed b=Complex(1.0,t); Lentz s method ( 5.

2). c=Complex(BIG,0.0); d=h=1.

0/b; for (i=1;i<MAXIT;i++) { a= -i*i; b += 2.0; d=1.0/(a*d+b); Denominators cannot be zero.

c=b+a/c; del=c*d; h *= del; if (abs(real(del)-1.0)+abs(imag(del)) <= EPS) break; } if (i >= MAXIT) throw("cf failed in cisi"); h=Complex(cos(t),-sin(t))*h; cs= -conj(h)+Complex(0.0,PIBY2); } else { Evaluate both series simultaneously.

cisi.h.
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