Hi Rooters,
I was working on some integrals lately and I stumbled on "exponential
integrals". I was surprised that they were not implemented in TMath.
Anyway, I decided to look them up from the online Numerical Recipes in
C. I simply copied the codes for E_{n}(x) and Ei(x), where x>0. By the
way,
E_{n}(x) = \int_{1}^{infty} { exp(-xt)/t^{n} } dt and
Ei(x) = -\int_{-x}^{infty} { exp(-t)/t } dt
However, since I needed to input negative values for Ei(x), I used the
fact that
E_{1}(x) = -Ei(-x)
and placed it into the code.
Anyway, here's the code, just in case someone else will need them like I
did.
//************** EXPONENTIAL INTEGRAL Ei ******
// define: ei(x) = -\int_{-x}^{\infty}{exp(-t)/t}dt, for x>0
// power series: ei(x) = eulerconst + ln(x) + x/(1*1!) + x^2/(2*2!) + ...
double ei(double x)
{ // taken from Numerical Recipes in C
const double euler = 0.57721566; // Euler's constant, gamma
const int maxit = 100; // max. no. of iterations allowed
const double fpmin = 1.0e-40; // close to smallest floating-point
number
const double eps = 1.0e-30; // relative error, or absolute error
near
// the zero of Ei at x=0.3725
// I actually changed fpmin and eps into smaller values than in NR
int k;
double fact, prev, sum, term;
// special case
if(x < 0) return -expint(1,-x);
if(x == 0.0) { cout << "Bad argument for ei(x)" << endl; return -1; }
if(x < fpmin) return log(x)+euler;
if(x <= -log(eps)) {
sum = 0;
fact = 1;
for(k=1; k<=maxit; k++) {
fact *= x/k;
term = fact/k;
sum += term;
if(term < eps*sum) break;
}
if(k>maxit) { cout << "Series failed in ei(x)" << endl; return -1; }
return sum+log(x)+euler;
} else {
sum = 0;
term = 1;
for(k=1; k<=maxit; k++) {
prev = term;
term *= k/x;
if(term<eps) break;
if(term<prev) sum+=term;
else {
sum -= prev;
break;
}
}
return exp(x)*(1.0+sum)/x;
}
}
//*********************************************
//************** EXPONENTIAL INTEGRALS En *****
// define: E_n(x) = \int_1^infty{exp(-xt)/t^n}dt, x>0, n=0,1,...
double expint(int n, double x) {
// based on Numerical Recipes in C
const double euler = 0.57721566; // Euler's constant, gamma
const int maxit = 100; // max. no. of iterations allowed
const double fpmin = 1.0e-30; // close to smallest floating-point
number
const double eps = 6.0e-8; // relative error, or absolute error near
// the zero of Ei at x=0.3725
int i, ii, nm1;
double a,b,c,d,del,fact,h,psi,ans;
nm1=n-1;
if(n<0 || x<0 || (x==0 && (n==0 || n==1))) {
cout << "Bad argument for expint(n,x)" << endl; return -1;
}
else {
if(n==0) ans=exp(-x)/x;
else {
if(x==0) ans=1.0/nm1;
else {
if(x>1) {
b=x+n;
c=1.0/fpmin;
d=1.0/b;
h=d;
for(i=1; i<maxit; i++) {
a = -i*(nm1+i);
b += 2.0;
d=1.0/(a*d+b);
c=b+a/c;
del=c*d;
h *= del;
if(fabs(del-1.0)<eps) {
ans=h*exp(-x);
return ans;
}
}
cout << "***continued fraction failed in expint(n,x)!!!" << endl;
return -1;
} else {
ans = (nm1!=0 ? 1.0/nm1 : -log(x)-euler);
fact=1;
for(i=1; i<=maxit; i++) {
fact *= -x/i;
if(i!=nm1) del = -fact/(i-nm1);
else {
psi = -euler;
for(ii=1; ii<=nm1; ii++) psi += 1.0/ii;
del = fact*(-log(x)+psi);
}
ans += del;
if(fabs(del)<fabs(ans)*eps) return ans;
}
cout << "***series failed in expint!!!" << endl;
return -1;
}
}
}
}
return ans;
}
//*********************************************
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