Logarithmic form

Any formula written in terms of logarithms may be said to be in logarithmic form.

Logarithmic differential forms

In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a 1-form that, locally at least, can be written

:$frac\{df\}\{f\}$

for some meromorphic function (resp. rational function) "f". That is, for some open covering, there are local representations of this differential form as a logarithmic derivative (modified slightly with the exterior derivative "d" in place of the usual differential operator "D"). These forms are quite highly constrained in their behaviour. For example on a Riemann surface it follows that they have simple poles, and everywhere integer residues at them. In higher dimension one needs the Poincaré residue to formulate their distinctive behaviour at places where "f" takes the value 0 or ∞.

Classically, for example in elliptic function theory, the logarithmic differential forms were recognised as complementary to the differentials of the first kind. They were sometimes called "differentials of the second kind" (and, with an unfortunate inconsistency, also sometimes "of the third kind"). The classical theory has now been subsumed as an aspect of Hodge theory. For a Riemann surface "S", for example, the differentials of the first kind account for the term "H"^0,1 in "H"¹("S"), when by the Dolbeault isomorphism it is interpreted as the sheaf cohomology group "H"⁰("S",Ω); this is tautologous considering their definition. The "H"^1,0 direct summand in "H"¹("S"), as well as being interpreted as "H"¹("S",O) where O is the sheaf of holomorphic functions on "S", can be identified more concretely with a vector space of logarithmic differentials.

Number theory

See Linear forms in logarithms.

External links

* [http://www.ucl.ac.uk/Mathematics/geomath/level2/hyper/hy8d.html The logarithmic form for inverse hyperbolics]
* [http://www.intmath.com/MethInt/2_BLog.php Ngee Ann Polytechnic: Methods of Integration: The Basic Logarithm Form]