Logarithmic integral function: Difference between revisions
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: π(''n'') ~ Li(n) = ∫<sub>''2''</sub><sup>''n''</sup> 1/ ln ''t'' d''t''. |
: π(''n'') ~ Li(n) = ∫<sub>''2''</sub><sup>''n''</sup> 1/ ln ''t'' d''t''. |
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This integral is in a connection with ''integral exponential function'' such as that li(''x'') = Ei (ln ''x''). If we substitute ''x'' with e<sup>''u''</sup>, we get a |
This integral is in a connection with ''integral exponential function'' such as that li(''x'') = Ei (ln ''x''). If we substitute ''x'' with e<sup>''u''</sup>, we get a series: |
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:li(e<sup>''u''</sup>) = γ + ln ''u'' + ''u'' + ''u''<sup>2</sup>/2 · 2! + ''u''<sup>3</sup>/3 · 3! + ''u''<sup>4</sup>/4 · 4! - ..., |
:li(e<sup>''u''</sup>) = γ + ln ''u'' + ''u'' + ''u''<sup>2</sup>/2 · 2! + ''u''<sup>3</sup>/3 · 3! + ''u''<sup>4</sup>/4 · 4! - ..., |
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Revision as of 17:00, 1 April 2002
Definite integral defined as:
- ∫0x 1/ln t dt
is a non-elemental function called logarithmic integral or integral logarithm and denoted with li(x) or Li(x). For x > 1 in a point t=1 this integral diverges, in this case we use for Li(x) the main value of unessential integral. Logarithmic integral with the main value of nondefinite integral comes in a variety of formulas concerning the density of primes in number theory and specially in prime numbers theorem, where for example the estimation for prime counting function π(n) is:
- π(n) ~ Li(n) = ∫2n 1/ ln t dt.
This integral is in a connection with integral exponential function such as that li(x) = Ei (ln x). If we substitute x with eu, we get a series:
- li(eu) = γ + ln u + u + u2/2 · 2! + u3/3 · 3! + u4/4 · 4! - ...,
where γ ≈ 0.57721 56649 01532 is Euler-Mascheroni's constant.