Exponential Integrals

Table of Available Exponential Integrals

• Exponential Integral - Ei(x) =
  Entire Exponential Integral - Ein(x) = ∫_-∞^x ( 1 - e ^-t ) / t dt
  Schlömilch's Exponential Integrals - En(x) = ∫₁^∞ ( e ^-xt / t ⁿ ) dt
  Alpha Exponential Integral of order n - α_n(x) = ∫₁^∞ tⁿ e ^-xt dt
  Beta Exponential Integral of order n - β_n(x) = ∫_-1¹ tⁿ e ^-xt dt

  Exponential Integral

  The exponential integral, Ei, is defined as defined as
  

  ∫	x	( e t / t ) dt
  	-∞

  Ei(x) = ∫-∞x ( e t / t ) dt.

  Ei(x) is strictly negative for x < 0 and approaches 0 as x → -∞. The integral becomes negatively infinite at x = 0. For x > 0 the integral is evaluated using Cauchy's principle value:
  

  Ei(x) = ∫-∞x ( e t / t ) dt = limε → 0+[ ∫-∞-ε ( e t / t ) dt + ∫ εx ( e t / t ) dt].

  Ei(x) becomes postively infinite as x → ∞.
  

  Function List

  Functions:
  
  • double Exponential_Integral_Ei( double x )
     
    This function returns Ei(x) for x ≠ 0 and if x = 0, -DBL_MAX is returned.
  • long double xExponential_Integral_Ei( long double x )
     
    This function is the same as Exponential_Integral_Ei except that the argument has type long double and returns a long double.
  C source code is available for these routines:
  • exponential_integral_Ei.c

  Entire Exponential Integral

  The entire exponential integral is defined as
  

  Ein(x) = ∫-∞x ( 1 - e -t ) / t dt.

  In terms of the exponential integral, Ei(x)

  Ein(x) = γ + ln |x| - Ei(-x),

  where γ is Euler's constant

  γ = limn→∞[&Sigmam = 1∞ (1 / m) - ln n].

  Ein(x) is strictly negative for x < 0, Ein(x) = 0 at x = 0, and strictly positive for x > 0.
  

  Function List

  Functions:
  
  • double Exponential_Integral_Ein( double x )
     
    This function returns Ein(x).
  C source code is available for this routine:
  • exponential_integral_Ein.c

  Schlömilch's Exponential Integrals

  Schlömilch's exponential integral of order n, n a nonnegative integer, is defined as
  

  En(x) = ∫1∞ ( e -xt / t n ) dt, where x > 0.

  The Schlömilch's exponential integral of order 0 is readily integrated
  

  E0(x) = ∫1∞ e -xt dt. = e -x / x.

  The Schlömilch's exponential integral of order 1 is readily expressed in terms of the exponential integral Ei
  

  E1(x) = ∫1∞ e -xt / t dt = -Ei(-x).

  If x < 0, then the integrals diverge and if x = 0, then for n = 0 or 1, the integrals diverge and if n ≥ 2 , then the integral E_n(0) = 1 / (n - 1).
  

  Function List

  Functions:
  
  • double Exponential_Integral_En( double x, int n )
     
    This function returns En(x) if x > 0. If x < 0, then DBL_MAX is returned. If x = 0, then if n = 0 or 1, then DBL_MAX is returned, otherwise for postive n ≥ 2 then 1 / (n - 1) is returned.
    Note that the order n must be nonnegative.

  C source code is available for this routine:
  • exponential_integral_En.c

  Alpha Exponential Integral of order n

  The alpha exponential integral of order n is defined as
  

  αn(x) = ∫1∞ tn e -xt dt

  for n = 0,1,2,... and x > 0.
  

  Function List

  Functions:
  
  • double Exponential_Integral_Alpha_n( double x, int n )
     
    This function returns α_n(x) if x > 0 and returns DBL_MAX if x ≤ 0, where the order n is a nonnegative integer.
  C source code is available for this routine:
  • exponential_integral_alpha_n.c

  Beta Exponential Integral of order n

  The beta exponential integral of order n is defined as
  

  βn(x) = ∫-11 tn e -xt dt

  for n = 0,1,2,... and x ∈ R.

  Function List

  Functions:
  
  • double Exponential_Integral_Beta_n( double x, int n )
     
    This function returns β_n(x), where the order n is a nonnegative integer.

  C source code is available for these routines:
  • exponential_integral_beta_n.c