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{{r|Complex analysis}}

...cially on [[complex analysis]]. But it is by no means necessary to rely on complex analysis here. A proof using [[field theory]] is alluded to at the very end of this

* Ahlfors, Lars V. (1953). Complex analysis. McGraw-Hill Book Company, Inc..

...ction|analytic]] for <math>\Re s \ge 1</math>, except for a simple [[Pole (complex analysis)|pole]] at <math>s=1</math> with residue 1. Then the [[Limit of a function|

From a theorem in [[complex analysis]], there is a unique analytic extension of this real function to the comple ...rigonometric functions become essential in the geometric interpretation of complex analysis. For example, with the above identity, if one considers the unit circle in

the study of [[Complex analysis|analytical]] objects (e.g., the [[Riemann zeta function]]) that encode prop ...nt of much of modern mathematics necessary for basic modern number theory: complex analysis, group theory, Galois theory -- accompanied by greater rigor in analysis an

...tically, despite its most important properties requiring only elementary [[complex analysis]]. An account of the function and its history that helped popularize it is

...nts due to division by zero; it is a [[meromorphic function]] with [[pole (complex analysis)|pole]]s at the nonpositive integers. The following image shows the graph o [[Karl Weierstrass]] further established the role of the gamma function in [[complex analysis]], starting from yet another product representation,

This concept regards functions that have [[Pole (complex analysis)|poles]]&mdash;isolated singularities, i.e., points where a function goes t

There are limits to what can be determined with portable equipment. For more complex analysis, either a transportable laboratory needs to be brought to the site, or, if

...between the input and output sides of the amplifier, the amplifier [[Pole (complex analysis)|pole]] lowest in frequency (usually an input pole) moves to a lower freque

..., one of the most fundamental open questions in mathematics, is drawn from complex analysis. [[Functional analysis]] focuses attention on (typically infinite-dimension

Internal evidence involves a more complex analysis of linguistic behaviour. For example, word [[stress (linguistics)|stress]]

Gienapp (1987) is the most complex analysis of the formation of the system. His has six basic findings. First, the real

...inite series) in number theory. Since he lived before the development of [[complex analysis]], most of his work is restricted is restricted to the formal manipulation