logarithmic derivative

logarithmic derivative (plural logarithmic derivatives)

1. (calculus, mathematical analysis) Given a real or complex function ${\displaystyle f}$ , the ratio of the value of the derivative to the value of the function, ${\displaystyle {\frac {f'}{f}}}$ , regarded as a function.
   • 1955, Frank S. Ham, The Quantum Defect Method, 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 ${\displaystyle U^{2}(r)}$ at the eigenvalue for any value of ${\displaystyle r}$ outside the core. If there exists a radius ${\displaystyle R}$ lying in the Coulomb region and sufficiently small that the logarithmic derivative at ${\displaystyle R}$ 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 ${\displaystyle (\ln \varphi (t))'=\varphi '(t)/\varphi (t)}$ , 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.

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

• Logarithmic differentiation on Wikipedia.Wikipedia
• Multiplicative calculus on Wikipedia.Wikipedia