Working definition
DLMF 6.2.1 defines the exponential integral E1 as the integral of e to the minus t over t from z to infinity, for non-zero z, with the path avoiding the negative real axis and a branch cut along the non-positive reals. 6.2.3 defines a companion, Ein, as the integral of one minus e to the minus t over t from zero to z, and records it as entire. The two are joined at 6.2.4: E1 equals Ein minus the logarithm of z minus Euler constant, which places the whole singular part in the logarithm rather than in the integral. 6.2.5 defines Ei as a principal value, 6.2.6 relates it to E1 on the negative axis, and 6.2.8 defines the logarithmic integral as a principal value equal to Ei of the logarithm. The sine and cosine integrals follow at 6.2.9 to 6.2.12, where Si and Cin are entire while Ci needs a principal value.
Notation
E1 of z and Ei of x for the exponential integralsEin for the entire companion and li for the logarithmic integralSi, si, Ci and Cin for the trigonometric integralsAssumptions
- E1 at 6.2.1 is stated for non-zero argument with the path avoiding the negative real axis, so it is branch-dependent rather than single valued.
- Ei at 6.2.5 and li at 6.2.8 are principal values, which are different objects from ordinary integrals.
- The relation at 6.2.4 holds with Euler constant as a stated term, so the constant is part of the definition rather than an approximation.
Invariants
- E1 defined with a branch cut along the non-positive reals, DLMF 6.2.1
- Ein is entire while E1 is not, DLMF 6.2.3
- E1 equals Ein minus log z minus Euler constant, DLMF 6.2.4
Reproducible procedure
- Choose E1 or Ei by which side of the axis the argument lies on, using the relation at 6.2.6 rather than continuing one across the cut.
- Where the argument is small, work through the entire companion at 6.2.3 and add the logarithmic term from 6.2.4, so the singularity is handled in closed form instead of by the quadrature.
- Treat li and Ci as principal values, and do not hand their integrands to a scheme that assumes an ordinary integral.
Error and boundary controls
- The singular behaviour near the origin sits in the logarithm by 6.2.4, so a scheme that integrates E1 directly near zero is fighting a singularity the identity already removes.
- A principal value is not an ordinary integral. A quadrature that ignores the interior pole in 6.2.5 or 6.2.8 returns a number without meaning rather than an inaccurate one, as with the third elliptic kind.
- The chapter gives definitions rather than algorithms and makes no accuracy claim about any evaluation.
What this does not establish
The chapter defines these integrals and their singular structure. It states no relation here between the logarithmic integral and the prime counting function, so none is claimed on this page.
Explicit applications
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This foundational concept currently supports related concepts; a direct domain bridge is scheduled for a later registry version.
Authoritative references
- [1]DLMF Chapter 6: Exponential, Logarithmic, Sine, and Cosine Integrals · National Institute of Standards and Technology
Establishes: Integral definitions for the exponential integrals with their branch cuts and principal values, the entire complementary form and the relation carrying Euler constant, the logarithmic integral, and the sine and cosine integrals with which of them are entire.
Boundary: The chapter defines the functions and states where each is singular or requires a principal value. It relates the logarithmic integral to the exponential integral and makes no claim here about approximating any counting function.
Direct answer
- DLMF 6.2.1 defines the exponential integral E1 as the integral of e to the minus t over t from z to infinity, for non-zero z, with the path avoiding the negative real axis and a branch cut along the non-positive reals. 6.2.3 defines a companion, Ein, as the integral of one minus e to the minus t over t from zero to z, and records it as entire. The two are joined at 6.2.4: E1 equals Ein minus the logarithm of z minus Euler constant, which places the whole singular part in the logarithm rather than in the integral. 6.2.5 defines Ei as a principal value, 6.2.6 relates it to E1 on the negative axis, and 6.2.8 defines the logarithmic integral as a principal value equal to Ei of the logarithm. The sine and cosine integrals follow at 6.2.9 to 6.2.12, where Si and Cin are entire while Ci needs a principal value.
Mechanism and method
- Choose E1 or Ei by which side of the axis the argument lies on, using the relation at 6.2.6 rather than continuing one across the cut.
- Where the argument is small, work through the entire companion at 6.2.3 and add the logarithmic term from 6.2.4, so the singularity is handled in closed form instead of by the quadrature.
- Treat li and Ci as principal values, and do not hand their integrands to a scheme that assumes an ordinary integral.
What is measured
- E1 defined with a branch cut along the non-positive reals, DLMF 6.2.1
- Ein is entire while E1 is not, DLMF 6.2.3
- E1 equals Ein minus log z minus Euler constant, DLMF 6.2.4
Limitations
- The singular behaviour near the origin sits in the logarithm by 6.2.4, so a scheme that integrates E1 directly near zero is fighting a singularity the identity already removes.
- A principal value is not an ordinary integral. A quadrature that ignores the interior pole in 6.2.5 or 6.2.8 returns a number without meaning rather than an inaccurate one, as with the third elliptic kind.
- The chapter gives definitions rather than algorithms and makes no accuracy claim about any evaluation.
- E1 at 6.2.1 is stated for non-zero argument with the path avoiding the negative real axis, so it is branch-dependent rather than single valued.
- Ei at 6.2.5 and li at 6.2.8 are principal values, which are different objects from ordinary integrals.
- The relation at 6.2.4 holds with Euler constant as a stated term, so the constant is part of the definition rather than an approximation.
What this does not establish
- The chapter defines these integrals and their singular structure. It states no relation here between the logarithmic integral and the prime counting function, so none is claimed on this page.