In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function is defined by the formula
<math display="block"> \frac{f'}{f} </math>
where is the derivative of . Intuitively, this is the infinitesimal relative change in ; that is, the infinitesimal absolute change in , namely scaled by the current value of .
When is a function of a real variable , and takes real, strictly positive values, this is equal to the derivative of , or the natural logarithm of . This follows directly from the chain rule:
In the field of Nevanlinna theory, an important lemma states that the proximity function of a logarithmic derivative is small with respect to the Nevanlinna characteristic of the original function, for instance <math>m(r,h'/h) = S(r,h) = o(T(r,h))</math>.
The multiplicative group
Behind the use of the logarithmic derivative lie two basic facts about GL<sub>1</sub>, that is, the multiplicative group of real numbers or other field. The differential operator
<math display="block"> X\frac{d}{dX} </math>
is invariant under dilation (replacing X by aX for a constant). And the differential form <math display="block">\frac{dx}{X}</math> is likewise invariant. For functions F into GL<sub>1</sub>, the formula
<math display="block">\frac{dF}{F}</math> is therefore a pullback of the invariant form.
Examples
- Exponential growth and exponential decay are processes with constant logarithmic derivative.
- In mathematical finance, the Greek λ is the logarithmic derivative of derivative price with respect to underlying price.
- In numerical analysis, the condition number is the infinitesimal relative change in the output for a relative change in the input, and is thus a ratio of logarithmic derivatives.
- The digamma function, and by extension the polygamma function, is defined in terms of the logarithmic derivative of the gamma function.
See also
- Elasticity of a function
- Product integral