Logarithmic derivative
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In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula
When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln(f), or the natural logarithm of f. This follows directly from the chain rule:[1][2]
Basic properties[edit]
Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does not take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have
A corollary to this is that the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function:
More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor:
Generalising in another direction, the logarithmic derivative of a power (with constant real exponent) is the product of the exponent and the logarithmic derivative of the base:
In summary, both derivatives and logarithms have a product rule, a reciprocal rule, a quotient rule, and a power rule (compare the list of logarithmic identities); each pair of rules is related through the logarithmic derivative.[2][verification needed]
Computing ordinary derivatives using logarithmic derivatives[edit]
Logarithmic derivatives can simplify the computation of derivatives requiring the product rule while producing the same result. The procedure is as follows: Suppose that and that we wish to compute . Instead of computing it directly as , we compute its logarithmic derivative. That is, we compute:
Multiplying through by ƒ computes f′:
This technique is most useful when ƒ is a product of a large number of factors. This technique makes it possible to compute f′ by computing the logarithmic derivative of each factor, summing, and multiplying by f.[2][verification needed]
For example, we can compute the logarithmic derivative of to be .[2]
Integrating factors[edit]
The logarithmic derivative idea is closely connected to the integrating factor method for first-order differential equations. In operator terms, write
In practice we are given an operator such as
Complex analysis[edit]
The formula as given can be applied more widely; for example if f(z) is a meromorphic function, it makes sense at all complex values of z at which f has neither a zero nor a pole. Further, at a zero or a pole the logarithmic derivative behaves in a way that is easily analysed in terms of the particular case
- zn
with n an integer, n ≠ 0. The logarithmic derivative is then
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 .[5][verification needed]
The multiplicative group[edit]
Behind the use of the logarithmic derivative lie two basic facts about GL1, that is, the multiplicative group of real numbers or other field. The differential operator
Examples[edit]
- Exponential growth and exponential decay are processes with constant logarithmic derivative.[citation needed]
- In mathematical finance, the Greek λ is the logarithmic derivative of derivative price with respect to underlying price.[citation needed]
- 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.[citation needed]
See also[edit]
- Generalizations of the derivative – Fundamental construction of differential calculus
- Logarithmic differentiation – Method of mathematical differentiation
- Elasticity of a function
References[edit]
- ^ a b "Logarithmic derivative - Encyclopedia of Mathematics". encyclopediaofmath.org. 7 December 2012. Retrieved 12 August 2021.
{{cite web}}
: CS1 maint: url-status (link) - ^ a b c d e f g "logarithmic derivative". planetmath.org. Retrieved 2021-08-12.
- ^ Gonzalez, Mario (1991-09-24). Classical Complex Analysis. CRC Press. ISBN 978-0-8247-8415-7.
- ^ "Logarithmic residue - Encyclopedia of Mathematics". encyclopediaofmath.org. 7 June 2020. Retrieved 2021-08-12.
{{cite web}}
: CS1 maint: url-status (link) - ^ Zhang, Guan-hou (1993-01-01). Theory of Entire and Meromorphic Functions: Deficient and Asymptotic Values and Singular Directions. American Mathematical Soc. p. 18. ISBN 978-0-8218-8764-6. Retrieved 12 August 2021.