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LogIntegral

LogIntegral[z]
is the logarithmic integral function .
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The logarithmic integral function is defined by , where the principal value of the integral is taken.
  • LogIntegral[z] has a branch cut discontinuity in the complex z plane running from to .
  • For certain special arguments, LogIntegral automatically evaluates to exact values.
  • LogIntegral can be evaluated to arbitrary numerical precision.
Evaluate numerically:
Series expansion around the branch point at :
Evaluate numerically:
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Series expansion around the branch point at :
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Click for copyable input
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Evaluate to high precision:
The precision of the output tracks the precision of the input:
LogIntegral threads element-wise over lists:
LogIntegral can take complex number inputs:
Simple exact values are generated automatically:
Series expansions on either side of the branch point at :
TraditionalForm formatting:
LogIntegral can be applied to power series:
Approximate number of primes less than :
Compare with exact counts:
Plot the real part in the complex plane:
Plot the absolute value in the complex plane:
Find an approximation to Soldner's constant :
Use FullSimplify to simplify expressions into logarithmic integrals:
Use FunctionExpand to write expressions in logarithmic integrals when possible:
Find the numerical root:
Obtain LogIntegral from integrals and sums:
Integrals:
In traditional form, parentheses are needed around the argument:
Nested integrals:
Plot the Riemann surface of LogIntegral:
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