Offset Log Integral Li Calculator
Approximate offset logarithmic integral Li(x)=∫2^x dt/log t by caller even-panel Simpson rule; no principal-value li(x), singularity crossing or automatic quadrature error bound.
Description
The offset logarithmic integral starts at 2, whereas the principal-value li function uses a different convention.
The offset logarithmic integral starts at 2, whereas the principal-value li function uses a different convention. The singularity at 1 cannot be crossed by this real quadrature.1
Approximate offset logarithmic integral Li(x)=∫2^x dt/log t by caller even-panel Simpson rule; no principal-value li(x), singularity crossing or automatic quadrature error bound.
When to use Offset Log Integral Li Calculator
Use this tool to approximate offset logarithmic integral Li(x)=∫2^x dt/log t by caller even-panel Simpson rule; no principal-value li(x), singularity crossing or automatic quadrature error bound.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Offset Log Integral Li Calculator and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter x, simpsonPanels using the field table and calculation rules, then run the tool.
- Read offsetLogarithmicIntegral. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| x | X | number | Required |
| simpsonPanels | Simpson panels | number | Required |
Use 1<x≤one million and an even Simpson panel count from 2 to 10,000. Integrate over ordered endpoints and reverse the sign when x<2. x=2 validly returns zero.
Li(x) = integral from 2 to x of 1/log(t) dt Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| offsetLogarithmicIntegral | Offset logarithmic integral | number |
{
"input": {
"x": 4,
"simpsonPanels": 100
},
"output": {
"offsetLogarithmicIntegral": 1.9224213202006408
}
}Limitations and assumptions
No principal-value continuation or automatic quadrature error bound is supplied. Resolution near x=1 depends on the chosen panels.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Elliptic Integral First Kind Approximation: Approximate complete real elliptic integral of the first kind on [0,π/2] with caller even-panel Simpson rule; parameter m=k² explicitly differs from modulus k, no hidden error guarantee near singular m=1. For a complementary task, use Dirichlet Eta Approximation: Compute finite real Dirichlet eta series for s>0 and analytic alternating-tail bound; caller exponent changes function, unlike Catalan constant fixed odd-square series.
References
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Offset Log Integral Li Calculator — tool.io
Similar or alternative tools
- Elliptic Integral First Kind Approximation
Approximate complete real elliptic integral of the first kind on [0,π/2] with caller even-panel Simpson rule; parameter m=k² explicitly differs from modulus k, no hidden error guarantee near singular m=1.
- Dirichlet Eta Approximation
Compute finite real Dirichlet eta series for s>0 and analytic alternating-tail bound; caller exponent changes function, unlike Catalan constant fixed odd-square series.