logarithmic derivative

English

Pronunciation

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Noun

logarithmic derivative (plural logarithmic derivatives)

1. (calculus, mathematical analysis) Given a real or complex function $f$, the ratio of the value of the derivative to the value of the function, ${\frac {f'}{f}}$, regarded as a function.
   • 1955, Frank S. Ham, “The Quantum Defect Method”, in Frederick Seitz, David Turnbull, editors, Solid State Physics, Volume 1, Academic Press, page 138:

     From this Coulomb function, we can then calculate the logarithmic derivative of $U^{2}(r)$ at the eigenvalue for any value of $r$ outside the core. If there exists a radius $R$ lying in the Coulomb region and sufficiently small that the logarithmic derivative at $R$ is a smooth function of the energy, we can obtain the logarithmic derivative for arbitrary energies within a reasonable range by interpolating it between the eigenvalues.

   • 1994, Bjarne S. Jensen, The Dynamic Systems of Basic Economic Growth Models, Kluwer Academic, page 329:

     In economics, it is popular to consider the logarithmic derivative $(\ln \varphi (t))'=\varphi '(t)/\varphi (t)$, as a convenient growth measure.

   • 1995, Philip G. Burke, Charles J. Joachain, Theory of Electron-Atom Collisions: Part 1: Potential Scattering, Plenum Press, page 86:

     In this section we consider the R-matrix method in which a solution is first found in an inner region 0 ≤ r ≤ a by expanding in a basis set yielding the logarithmic derivative of the wave function on the boundary r = a.

Usage notes

• The logarithmic derivative can be interpreted intuitively as the infinitesimal relative change in $f$ at any given point.
• If $f(x)$ is a differentiable function of a real variable and takes only positive values (so that $\ln f(x)$ is defined), the chain rule applies and the logarithmic derivative is equal to the derivative of the logarithm: $\textstyle {\frac {f'(x)}{f(x)}}=\left(\ln f(x)\right)'$.
• The definition above is more broadly applicable: for $f(z)$ a function of a complex variable, its logarithmic derivative will be computable so long as $f(z)\neq 0$ and $f'(z)$ is defined.

Translations

ratio of the value of the derivative to the value of the function, regarded as a function

Further reading

• Logarithmic differentiation on Wikipedia.Wikipedia