Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

HarmonicNumber[n,a] for symbolic a:

HarmonicNumber[n,a] for symbolic n:

Value at zero:

Find a value of n for which HarmonicNumber[n]=1.5:

Plot the HarmonicNumber function:

Plot the HarmonicNumber function for various orders:

Plot the real part of HarmonicNumber:

Plot the imaginary part of HarmonicNumber:

Real domain of HarmonicNumber:

Complex domain:

Real range of HarmonicNumber:

HarmonicNumber threads elementwise over lists and arrays:

HarmonicNumber is not an analytic function:

However, it is meromorphic:

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:

TraditionalForm formatting:

First derivative with respect to n:

Higher derivatives with respect to n:

Plot the higher derivatives with respect to n:

Formula for the derivative with respect to n:

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

HarmonicNumber is defined through the identity:

Recurrence identities:

Series expansion at infinity:

HarmonicNumber can be applied to power series:

Evaluate at exact arguments:

Series expansion at any point:

Series expansion at infinity: