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{{For|essential singularities of real valued functions|Classification of discontinuities}}
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[[File:Essential singularity.png|right|220px|thumb|Plot of the function exp(1/''z''), centered on the essential singularity at ''z''=0. The hue represents the [[Arg (mathematics)|complex argument]], the luminance represents the [[absolute value]]. This plot shows how approaching the essential singularity from different directions yields different behaviors (as opposed to a pole, which, approached from any direction, would be uniformly white).]]
[[File:Modell des Graphen von 6w=eˆ(1-6z) -Schilling XIV, 6 - 312- (2).jpg|thumb|Model illustrating essential singularity of a complex function 6w=exp(1/(6z))]]
 
In [[complex analysis]], an '''essential singularity''' of a function is a "severe" [[singularity (mathematics)|singularity]] near which the function exhibits extreme behavior.  
 
The category ''essential singularity'' is a "left-over" or default group of singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner &ndash; [[removable singularity|removable singularities]] and [[pole (complex analysis)|pole]]s.
 
==Formal description==
Consider an [[open set|open subset]] ''U'' of the [[complex plane]] '''C'''. Let ''a'' be an element of ''U'', and ''f''&nbsp;:&nbsp;''U''&nbsp;\&nbsp;{''a''}&nbsp;→&nbsp;'''C''' a [[meromorphic function]]. The point ''a'' is called an ''essential singularity'' of the function ''f'' if the singularity is neither a [[pole (complex analysis)|pole]] nor a [[removable singularity]].
 
For example, the function ''f''(''z'') = ''e''<sup>1/''z''</sup> has an essential singularity at ''z'' = 0.
 
==Alternate descriptions==
 
Let ''a'' be a complex number, assume that ''f''(''z'') is not defined at ''a'' but is [[Analytic function|analytic]] in some region ''U'' of the complex plane, and that every [[Open set|open]] [[neighbourhood (mathematics)|neighbourhood]] of ''a'' has non-empty intersection with ''U''.  
 
If both
:<math>\lim_{z \to a}f(z)</math> &nbsp; and &nbsp; <math>\lim_{z \to a}\frac{1}{f(z)}</math> &nbsp; exist, then ''a'' is a [[removable singularity]] of both ''f'' and 1/''f''.  
 
If
:<math>\lim_{z \to a}f(z)</math> &nbsp; exists but &nbsp; <math>\lim_{z \to a}\frac{1}{f(z)}</math> &nbsp; does not exist, then ''a'' is a [[zero (complex analysis)|zero]] of ''f'' and a [[pole (complex analysis)|pole]]  of 1/''f''.
 
Similarly, if
:<math>\lim_{z \to a}f(z)</math> &nbsp; does not exist but &nbsp; <math>\lim_{z \to a}\frac{1}{f(z)}</math> &nbsp; exist, then ''a'' is a pole of ''f'' and a zero of 1/''f''.
 
If neither
:<math>\lim_{z \to a}f(z)</math> &nbsp;  nor &nbsp; <math>\lim_{z \to a}\frac{1}{f(z)}</math> &nbsp; exists, then ''a'' is an essential singularity of both ''f'' and 1/''f''.
 
Another way to characterize an essential singularity is that the [[Laurent series]] of ''f'' at the point ''a'' has infinitely many negative degree terms (i.e., the [[principal part]] of the Laurent series is an infinite sum).
 
The behavior of holomorphic functions near their essential singularities is described by the [[Casorati–Weierstrass theorem]] and by the considerably stronger [[Picard's great theorem]]. The latter says that in every neighborhood of an essential singularity ''a'', the function ''f'' takes on ''every'' complex value, except possibly one, infinitely many times.
 
==References==
{{refbegin}}
*{{citeweb|url=http://mathworld.wolfram.com/EssentialSingularity.html|title=Essential Singularity at Mathworld|accessdate=18 February 2008}}
*Lars V. Ahlfors; ''Complex Analysis'', McGraw-Hill, 1979
*Rajendra Kumar Jain, S. R. K. Iyengar; ''Advanced Engineering Mathematics''. Page 920. Alpha Science International, Limited, 2004. ISBN 1-84265-185-4
{{refend}}
 
== External links ==
* '' [http://demonstrations.wolfram.com/AnEssentialSingularity/ An Essential Singularity]'' by [[Stephen Wolfram]], [[Wolfram Demonstrations Project]].
* [http://planetmath.org/encyclopedia/EssentialSingularity.html Essential Singularity on Planet Math]
*[http://www.encyclopediaofmath.org/index.php/Essential_singular_point Essential singularity] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
 
[[Category:Complex analysis]]

Revision as of 03:18, 6 February 2014

The ac1st16.dll error is annoying and fairly popular with all types of Windows computers. Not only does it make your computer run slower, nevertheless it can additionally avoid we from using a variety of programs, including AutoCAD. To fix this issue, you need to utilize a simple way to remedy all possible difficulties which cause it. Here's what we have to do...

We have to know several effortless and cheap ways which will solve the issue of your computer plus speed it up. The earlier we fix it, the less damage your computer gets. I might tell about certain valuable techniques that usually assist you to accelerate we computer.

One of the many overlooked reasons a computer might slow down is considering the registry has become corrupt. The registry is essentially your computer's operating program. Anytime you're running a computer, entries are being made plus deleted from a registry. The effect this has is it leaves false entries inside a registry. So, a computer's resources must function about these false entries.

The 1328 error is a widespread issue caused by your program being unable to properly procedure various changes for a system or Microsoft Office. If you have this error, it generally signifies which the computer is either unable to read the actual update file or a computer has issues with all the settings it's utilizing to run. To fix this problem, we first have to change / fix any difficulties that the computer has with its update files, and then repair some of the issues which a system can have.

The 2nd step to fixing these mistakes is to utilize a system called a "fix it utilities" to scan through the computer and fix any of the registry errors that might equally be leading for this error. A registry cleaner is a software system which can scan through the computer plus repair any of the difficulties which Windows has inside, allowing the computer to "remember" all of the settings it has when it loads up. Although the registry is continually being utilized to help load up a large amount of programs on the PC, it's continually being saved incorrectly - leading to a large amount of errors to be formed. To fix this issue, it's suggested we download a registry cleaner from the Internet and install it on a Pc, permitting Windows to run smoother again.

Another element is registry. It is one of the most crucial part in a Windows XP, Vista running systems. When Windows start up, it read related information from registry and load into computer RAM. This takes up a big part of the startup time. After the data is all loaded, computer runs the business programs.

It is critical which we remove obsolete registry entries from the program on a regular basis, if you would like your system to run quicker, that is. If you don't keep the registry clean, a time comes when a program will stop working completely. Then, the only option would be to reformat a hard drive and begin over!

Registry products can assist the computer run inside a better mode. Registry cleaners ought to be part of the usual scheduled maintenance system for your computer. You don't have to wait forever for a computer or the programs to load plus run. A little maintenance may bring back the speed we lost.

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