Information AboutDerivative Of The Logarithm |
| CATEGORIES ABOUT LOGARITHMIC DERIVATIVE | |
| differential calculus | |
| complex analysis | |
f where ''f′'' is the Derivative of ''f''. When ''f'' is a function ''f(x)'' of a real variable ''x'', and takes real, ''strictly Positive '' values, this is indeed the formula for (log ''f'')′, that is, the derivative of the Natural Logarithm of ''f'', as follows from the Chain Rule . FORMULAE Some basic calculus applications of the formula :(ln(''uv''))′ = (''uv'')′/(''uv'') = (''u''′)/''u'' + (''v''′)/''v'' that expresses the way the ''logarithmic derivative of a product'' is the ''sum of the logarithmic derivatives of the factors''. One consequence is the conventional Leibniz Rule :(''uv'')′ = ''u''′''v'' + ''uv''′ which follows by clearing denominators. Another is the Quotient Rule : :(ln(''u''/''v''))′ = (''u''′)/''u'' − (''v''′)/''v'' ::= (''u′/v'')/(''u/v'') − (''uv′/v2'')/(''u/v'') ::= (''u′/v − uv′/v2'')/(''u/v'') ::= ((''u′v − uv′'')''/v2'')/(''u/v'') ::= (''u/v'')′/(''u/v'') :so (''u/v'')′ = (''u′v − uv′'')/''v2''. INTEGRATING FACTORS The logarithmic derivative idea is closely connected to the Integrating Factor method, for First Order Differential Equations . In Operator terms, write D and let ''M'' denote the operator of multiplication by some given function ''G''(''x''). Then M can be written (by the Product Rule ) as
G In practice we are given an operator such as D and wish to solve equations L for the function ''h'', given ''f''. This then reduces to solving G which has as solution :exp(∫''F'') with any Indefinite Integral of ''F''. COMPLEX ANALYSIS 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 n 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 . THE MULTIPLICATIVE GROUP Behind the use of the logarithmic derivative lie two basic facts about ''GL1'', that is, the multiplicative group of Real Number s or other Field . The Differential Operator X-1d/dX is Invariant under 'translation' (replacing ''X'' by ''aX'' for ''a'' constant). And the Differential Form dX/X is likewise invariant. For functions ''F'' into ''GL1'', the formula dF/F is therefore a '' Pullback '' of the invariant form. |
|
|