gives the n harmonic number .
gives the harmonic number of order r.
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The harmonic numbers are given by with .
- HarmonicNumber can be evaluated to arbitrary numerical precision.
- HarmonicNumber automatically threads over lists.
- HarmonicNumber can be used with Interval and CenteredInterval objects. »
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Basic Examples (7)
Series expansion at Infinity:
Numerical Evaluation (5)
Specific Values (4)
Function Properties (11)
Real domain of HarmonicNumber:
Real range of HarmonicNumber:
HarmonicNumber threads elementwise over lists and arrays:
HarmonicNumber is not an analytic function:
HarmonicNumber is neither non-increasing nor non-decreasing:
HarmonicNumber is not injective:
HarmonicNumber is surjective:
HarmonicNumber is neither non-negative nor non-positive:
HarmonicNumber has both singularities and discontinuities for negative integers:
HarmonicNumber is neither convex nor concave:
Compute the indefinite integral using Integrate:
Series Expansions (2)
Function Identities and Simplifications (2)
HarmonicNumber is defined through the identity:
Generalizations & Extensions (5)
Harmonic Numbers (2)
HarmonicNumber can be applied to power series:
Picking the best candidate out of n after x evaluated choices [more info]:
Properties & Relations (9)
Use FullSimplify to simplify expressions containing harmonic numbers:
General term in the series expansion of HarmonicNumber:
The generating function for HarmonicNumber:
The exponential generating function for HarmonicNumber:
Wolfram Research (1999), HarmonicNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/HarmonicNumber.html.
Wolfram Language. 1999. "HarmonicNumber." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HarmonicNumber.html.
Wolfram Language. (1999). HarmonicNumber. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HarmonicNumber.html