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 <math display="block"> \frac{f'}{f} </math> where <math>f'</math> is the derivative of f.<ref name=":0">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Intuitively, this is the infinitesimal relative change in f; that is, the infinitesimal absolute change in f, namely <math>f',</math> scaled by the current value of f.

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:<ref name=":0" /> <math display="block"> \frac{d}{dx}\ln f(x) = \frac{1}{f(x)} \frac{df(x)}{dx} </math>

Basic propertiesEdit

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 <math display="block"> (\log uv)' = (\log u + \log v)' = (\log u)' + (\log v)' . </math> So for positive-real-valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors. But we can also use the Leibniz law for the derivative of a product to get <math display="block"> \frac{(uv)'}{uv} = \frac{u'v + uv'}{uv} = \frac{u'}{u} + \frac{v'}{v} . </math> Thus, it is true for any function that the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors (when they are defined).

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: <math display="block"> \frac{(1/u)'}{1/u} = \frac{-u'/u^{2}}{1/u} = -\frac{u'}{u} , </math> just as the logarithm of the reciprocal of a positive real number is the negation of the logarithm of the number.Template:Citation needed

More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor: <math display="block"> \frac{(u/v)'}{u/v} = \frac{(u'v - uv')/v^{2}}{u/v} = \frac{u'}{u} - \frac{v'}{v} , </math> just as the logarithm of a quotient is the difference of the logarithms 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: <math display="block"> \frac{(u^{k})'}{u^{k}} = \frac {ku^{k-1}u'}{u^{k}} = k \frac{u'}{u} , </math> just as the logarithm of a power is the product of the exponent and the logarithm 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.

Computing ordinary derivatives using logarithmic derivativesEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Logarithmic derivatives can simplify the computation of derivatives requiring the product rule while producing the same result. The procedure is as follows: Suppose that Template:Nowrap and that we wish to compute <math>f'(x)</math>. Instead of computing it directly as Template:Nowrap, we compute its logarithmic derivative. That is, we compute: <math display="block">\frac{f'}{f} = \frac{u'}{u} + \frac{v'}{v}.</math>

Multiplying through by ƒ computes Template:Math: <math display="block">f' = f\cdot\left(\frac{u'}{u} + \frac{v'}{v}\right).</math>

This technique is most useful when ƒ is a product of a large number of factors. This technique makes it possible to compute Template:Math by computing the logarithmic derivative of each factor, summing, and multiplying by Template:Math.

For example, we can compute the logarithmic derivative of <math>e^{x^2}(x-2)^3(x-3)(x-1)^{-1}</math> to be <math>2x + \frac{3}{x-2} + \frac{1}{x-3} - \frac{1}{x-1}</math>.

Integrating factorsEdit

The logarithmic derivative idea is closely connected to the integrating factor method for first-order differential equations. In operator terms, write <math display="block"> D = \frac{d}{dx} </math> and let M denote the operator of multiplication by some given function G(x). Then <math display="block"> M^{-1} D M </math> can be written (by the product rule) as <math display="block">D + M^{*} </math> where <math> M^{*} </math> now denotes the multiplication operator by the logarithmic derivative <math display="block"> \frac{G'}{G}</math>

In practice we are given an operator such as <math display="block"> D + F = L </math> and wish to solve equations <math display="block"> L(h) = f </math> for the function h, given f. This then reduces to solving <math display="block"> \frac{G'}{G} = F </math> which has as solution <math display="block"> \exp \textstyle ( \int F ) </math> with any indefinite integral of F.Template:Citation needed

Complex analysisEdit

Template:See also 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

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with n an integer, Template:Math. The logarithmic derivative is then <math display="block">n/z</math> and one can draw the general conclusion that for f meromorphic, the singularities of the logarithmic derivative of f are all simple poles, with residue n from a zero of order n, residue −n from a pole of order n. See argument principle. This information is often exploited in contour integration.<ref>Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>Template:Verify source

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>.<ref>Template:Cite book</ref>Template:Verify source

The multiplicative groupEdit

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 <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 GL₁, the formula <math display="block">\frac{dF}{F}</math> is therefore a pullback of the invariant form.Template:Citation needed

ExamplesEdit

Exponential growth and exponential decay are processes with constant logarithmic derivative.Template:Citation needed
In mathematical finance, the Greek λ is the logarithmic derivative of derivative price with respect to underlying price.Template: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.Template:Citation needed
The digamma function, and by extension the polygamma function, is defined in terms of the logarithmic derivative of the gamma function.

ReferencesEdit

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