logarithmic derivative

 

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logarithmic derivative

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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”, 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 ${\displaystyle U^{2}(r)}$  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.

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