Trivial mathematical identities are relatively simple (for an experienced mathematician), though not necessarily unimportant.
Both of the above are derived from the following two equations that define a logarithm: (note that in this explanation, the variables of
Each of these logarithm properties correspond to their respective exponent law, and their derivations/proofs will hinge on those facts.
To state the logarithm of a power law formally: Derivation: Let
The left side of the equality can be simplified using a logarithm law, which states that
The following summation/subtraction rule is especially useful in probability theory when one is dealing with a sum of log-probabilities: Note that the subtraction identity is not defined if
The last limit is often summarized as "logarithms grow more slowly than any power or root of x".
In these cases, different representations[6] or methods must be used to evaluate the logarithm.
It is not uncommon in advanced mathematics, particularly in analytic number theory and asymptotic analysis, to encounter expressions involving differences or ratios of harmonic numbers at scaled indices.
[7] The identity involving the limiting difference between harmonic numbers at scaled indices and its relationship to the logarithmic function provides an intriguing example of how discrete sequences can asymptotically relate to continuous functions.
This identity is expressed as[8] which characterizes the behavior of harmonic numbers as they grow large.
in the limit) reflects how summation over increasing segments of the harmonic series exhibits integral properties, giving insight into the interplay between discrete and continuous analysis.
It also illustrates how understanding the behavior of sums and series at large scales can lead to insightful conclusions about their properties.
This result leverages the fact that the sum of the inverses of integers (i.e., harmonic numbers) can be closely approximated by the natural logarithm function, plus a constant, especially when extended over large intervals.
, precludes the possibility of the harmonic series approaching a finite limit, thus providing a clear mathematical articulation of its divergence.
This specific identity can be a consequence of these approximations, considering: The limit explores the growth of the harmonic numbers when indices are multiplied by a scaling factor and then differenced.
, the summation window encompasses an increasingly vast portion of the smallest possible terms of the harmonic series (those with astronomically large denominators), creating a discrete sum that stretches towards infinity, which mirrors how continuous integrals accumulate value across an infinitesimally fine partitioning of the domain.
For example: This method leverages the fine differences between closely related terms to stabilize the series.
ensures that adjustments are uniformly applied across all possible offsets within each block of
It helps to flatten out the discrepancies that might otherwise lead to divergent behavior in a straightforward harmonic series.
Note that the structure of the summands of this formula matches those of the interpolated harmonic number
Whereas the harmonic number difference computes the integral in a global sliding window, the double series, in parallel, computes the sum in a local sliding window—a shifting
trimmed from the series to establish the window's moving lower bound
Similarly, factorials can be approximated by summing the logarithms of the terms.
No single valued function on the complex plane can satisfy the normal rules for logarithms.
A single valued version, called the principal value of the logarithm, can be defined which is discontinuous on the negative x axis, and is equal to the multivalued version on a single branch cut.
The multiple valued version of log(z) is a set, but it is easier to write it without braces and using it in formulas follows obvious rules.
Principal value form: Multiple value forms: Where k1, k2 are any integers: As a consequence of the harmonic number difference, the natural logarithm is asymptotically approximated by a finite series difference,[8] representing a truncation of the integral at
Since the nth pronic number is asymptotically equivalent to the nth perfect square, it follows that: The prime number theorem provides the following asymptotic equivalence: where
These approximations extend to the real-valued domain through the interpolated harmonic number.