Differential of the first kind

<h2 id="differentials-of-the-second-and-third-kind">Differentials of the second and third kind</h2>
<p>The traditional terminology also included differentials of the second kind and of the third kind. The idea behind this has been supported by modern theories of algebraic differential forms, both from the side of more <a href="/facts/Hodge_theory/b6Mju74J">Hodge theory</a>, and through the use of morphisms to <a href="/facts/Commutative/WaYxJLxd">commutative</a> <a href="/facts/Algebraic_group/DDtBCIvh">algebraic groups</a>.
</p><p>The <a href="/facts/Weierstrass_zeta_function/88h4u2hI">Weierstrass zeta function</a> was called an <i>integral of the second kind</i> in <a href="/facts/Elliptic_function/mF03oI04">elliptic function</a> theory; it is a <a href="/facts/Logarithmic_derivative/VL7GIwkl">logarithmic derivative</a> of a <a href="/facts/Theta_function/g09xvoQQ">theta function</a>, and therefore has <a href="/facts/Simple_pole/gnA3U8VP">simple poles</a>, with integer residues. The decomposition of a (<a href="/facts/Meromorphic/LeNBd6Sh">meromorphic</a>) elliptic function into pieces of 'three kinds' parallels the representation as (i) a constant, plus (ii) a <a href="/facts/Linear_combination/CVjL6kuN">linear combination</a> of  translates of the Weierstrass zeta function, plus (iii) a function with arbitrary poles but no residues at them.
</p><p>The same type of decomposition exists in general, <i>mutatis mutandis</i>, though the terminology is not completely consistent. In the algebraic group (<a href="/facts/Generalized_Jacobian/X7oxVn6U">generalized Jacobian</a>) theory the three kinds are <a href="/facts/Abelian_varieties/dGSDVruG">abelian varieties</a>, algebraic tori, and <a href="/facts/Affine_space/tlvI3oX1">affine spaces</a>, and the decomposition is in terms of a <a href="/facts/Composition_series/3C7CHH3k">composition series</a>.
</p><p>On the other hand, a meromorphic abelian differential of the <i>second kind</i> has traditionally been one with residues at all poles being zero. One of the third kind is one where all poles are simple. There is a higher-dimensional analogue available, using the <a href="/facts/Poincar%C3%A9_residue/aECkV0X7">Poincaré residue</a>.
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<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Logarithmic_form/6Hhf0KmM">Logarithmic form</a></li></ul>

<ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Abelian_differential">"Abelian differential"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul>

Differential of the first kind open-in-new

Differential of the first kind