Heuristically this is because the logarithmic derivative of the Dedekind zeta-function of " K " always has a simple pole with residue & minus; 1 at " s " = 1.
22.
This integral function is extended by analytic continuation to all complex numbers except the non-positive integers ( where the function has simple poles ), yielding the meromorphic function we call the gamma function.
23.
The Weierstrass zeta function was called an " integral of the second kind " in elliptic function theory; it is a logarithmic derivative of a theta function, and therefore has simple poles, with integer residues.
24.
:The function e ^ {-z } only has a pole at infinity, and that is an essential singularity, who is saying it has a simple pole ? talk ) 12 : 03, 7 January 2014 ( UTC)
25.
Later, Riemann considered this function for complex values of " s " and showed that this function can be extended to a meromorphic function on the entire plane with a simple pole at " s " = 1.
26.
Menard, 55, a self-made captain of industry who amassed his fortune by parlaying a simple pole-barn building business in the Wisconsin dairyland into a chain of 105 home-improvement stores across the Midwest, harbored no grand illusions about Indy.
27.
However, if the second argument of \ scriptstyle { R _ C }, or the fourth argument, p, of \ scriptstyle { R _ J } is negative, then this results in a simple pole on the path of integration.
28.
The study of such algebraic solutions involves examining the topology of the deformation parameter space ( and in particular, its mapping class group ); for the case of simple poles, this amounts to the study of the action of braid groups.
29.
It s quite a break with the past, and, for this listener, both ill-fitting and a bad omen . " He remarked that the album's lyrical content " falls between the two simple poles of catfights over boys and sex-tease power trips over boys.
30.
By starting with the Weierstrass p-function and associating with it a group of doubly periodic functions with two simple poles, he was able to give a simple derivation of the Jacobian elliptic functions, as well as modifying the existing notation to provide a more systematic approach to the subject.
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