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[[File:Graph x squared undefined at x equals two.png|thumb|right|200px|A graph of a [[parabola]] with a '''removable singularity''' at ''x'' = 2]]
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In [[complex analysis]], a '''removable singularity''' (sometimes called a '''cosmetic singularity''') of a [[holomorphic function]] is a point at which the function is undefined, but it is possible to define the function at that point in such a way that the function is regular in a neighbourhood of that point.
 
For instance, the function
 
:<math> f(z) = \frac{\sin z}{z} </math>
 
has a singularity at ''z'' = 0. This singularity can be removed by defining ''f''(0) := 1, which is the [[Limit of a function|limit]] of ''f'' as ''z'' tends to 0. The resulting function is holomorphic. In this case the problem was caused by ''f'' being given an [[indeterminate form]]. Taking a power series expansion for <math>\frac{\sin(z)}{z}</math> shows that
:<math> f(z) = \frac{1}{z}\left(\sum_{k=0}^{\infty} \frac{(-1)^kz^{2k+1}}{(2k+1)!} \right) = \sum_{k=0}^{\infty} \frac{(-1)^kz^{2k}}{(2k+1)!} = 1 - \frac{z^2}{3!} + \frac{z^4}{5!} - \frac{z^6}{7!} + \cdots. </math>
 
Formally, if <math>U \subset \mathbb C</math> is an [[open subset]] of the [[complex plane]] <math>\mathbb C</math>, <math>a \in U</math> a point of <math>U</math>, and <math>f: U\setminus \{a\} \rightarrow \mathbb C</math> is a [[holomorphic function]], then <math>a</math> is called a '''removable singularity''' for <math>f</math> if there exists a holomorphic function <math>g: U \rightarrow \mathbb C</math> which coincides with <math>f</math> on <math>U\setminus \{a\}</math>. We say <math>f</math> is holomorphically extendable over <math>U</math> if such a <math>g</math> exists.
 
== Riemann's theorem ==
 
[[Bernhard Riemann|Riemann's]] theorem on removable singularities states when a singularity is removable:
 
''' Theorem.''' Let <math>D \subset C</math> be an open subset of the complex plane, <math>a \in D</math> a point of <math>D</math> and <math>f</math> a holomorphic function defined on the set <math>D \setminus \{a\}</math>. The following are equivalent:
 
# <math>f</math> is holomorphically extendable over <math>a</math>.  
# <math>f</math> is continuously extendable over <math>a</math>.  
# There exists a [[neighborhood (topology)|neighborhood]] of <math>a</math> on which <math>f</math> is [[bounded function|bounded]].
# <math>\lim_{z\to a}(z - a) f(z) = 0</math>.
 
The implications 1 ⇒ 2  ⇒ 3  ⇒ 4 are trivial. To prove 4 ⇒ 1, we first recall that the holomorphy of a function at <math>a</math> is equivalent to it being analytic at <math>a</math> ([[Proof that holomorphic functions are analytic|proof]]), i.e. having a power series representation. Define
 
:<math>
h(z) =
\begin{cases}
(z - a)^2 f(z) &  z \ne a ,\\
0              &  z = a .
\end{cases}
</math>
 
Clearly, ''h'' is holomorphic on ''D''&nbsp;\&nbsp;{''a''}, and there exists
:<math>h'(a)=\lim_{z\to a}\frac{(z - a)^2f(z)-0}{z-a}=\lim_{z\to a}(z - a) f(z)=0</math>
by 4, hence ''h'' is holomorphic on ''D'' and has a Taylor series about ''a'':
 
:<math>h(z) = c_0 + c_1(z-a) + c_2 (z - a)^2 + c_3 (z - a)^3 + \cdots \, .</math>
 
We have ''c''<sub>0</sub> = ''h''(''a'') = 0 and ''c''<sub>1</sub> = ''h{{'}}''(''a'') = 0; therefore
 
:<math>h(z) = c_2 (z - a)^2 + c_3 (z - a)^3 + \cdots \, .</math>
 
Hence, where z≠a, we have:
 
:<math>f(z)=h(z)/(z-a)^2 = c_2 + c_3 (z - a) + \cdots \, .</math>
 
However,
 
:<math>g(z) = c_2 + c_3 (z - a) + \cdots \, .</math>
 
is holomorphic on ''D'', thus an extension of ''f''.
 
== Other kinds of singularities ==
 
Unlike functions of a real variable, holomorphic functions are sufficiently rigid that their isolated singularities can be completely classified. A holomorphic function's singularity is either not really a singularity at all, i.e. a removable singularity, or one of the following two types:
 
#In light of Riemann's theorem, given a non-removable singularity, one might ask whether there exists a natural number <math>m</math> such that <math>\lim_{z \rightarrow a}(z-a)^{m+1}f(z)=0</math>. If so, <math>a</math> is called a '''[[pole (complex analysis)|pole]]''' of <math>f</math> and the smallest such <math>m</math> is the '''order''' of <math>a</math>. So removable singularities are precisely the [[pole (complex analysis)|pole]]s of order 0. A holomorphic function blows up uniformly near its poles.
#If an isolated singularity <math>a</math> of <math>f</math> is neither removable nor a pole, it is called an '''[[essential singularity]]'''. It can be shown that such an <math>f</math> maps every punctured open neighborhood <math>U \setminus \{a\}</math> to the entire complex plane, with the possible exception of at most one point.
 
==See also==
* [[Analytic capacity]]
* [[Removable discontinuity]]
 
== External links ==
{{Expand section|date=December 2009}}
 
[[Category:Analytic functions]]
[[Category:Meromorphic functions]]

Latest revision as of 17:57, 16 February 2014

Queen Elizabeth - the Virgin Queen, the state of Virginia is named soon after her. Virginia, one particular of the earliest British Settlements in America, is also the birthplace of none much less than eight US presidents. Aside from getting a state with fantastic historical value, it also gives one of the most scenic and pleasant areas to live in the county of Virginia Beach front.

The purchasers will check out out the garden, so make certain that you clear up your lawn before advertising and marketing it for sale. You have to mow the lawn, plant some bouquets, set potted plants, and eliminate litter. In doing so, you can appeal to customers will having attractive property.

Some issues that you can do to stop that: When you sign that deal, make it contingent on you finding a suitable tenant. We do that with every single sub-to now. Unless my tenant is prepared to go in, I don't close and consider the house. So just simply because they're ready to market you the home Arthur Falcone sub-to, doesn't make it a offer.

Repairs are likely to be the place you a good deal of negotiating. Inquire the operator about repairs and create them down. Stage out every single tiny factor you see that is not perfect. On every single product question him what it would expense to restore it. Attempt to get him to notify you a range of charges and publish down the optimum amount. Inquire him if he's certain the fix can be done for that amount often he'll increase the sum. Make confident to use the greatest volume in the relaxation of negotiations. When you begin adding up the fix volume this will aid reduced his value Arthur Falcone . Get a lot of notes about the repairs. The more you compose about them the far more it makes the owner informed of them.

You can commence your quest on the web. Log on to any look for engine and search for Virginia Beach real estate and you will uncover numerous web sites that offer several varieties of qualities, from ranches to colonial villas in many districts in Virginia Seaside. You can also purchase undeveloped land and then construct your personal house on it. So you see, there is no dearth of options in Virginia Beach Arthur Falcone.

This is 1 of the ideal things you can do prior to starting the property getting procedure, for a few of reasons. Very first of all, mortgage loan lenders like to see that you have some money reserves on hand. Next, you'll require income reserves for any unexpected costs or charges that may well arise (which is widespread).

3 A seventeen. Assesses can make only one Arthur Falcone charm towards the get of the section to an Appellate Authority (Bench of 11 users). Get of Appellate Authority ought to not be even more appealable. Appellate Tribunals ought to be set up in reasonable numbers. Users of tribunal should be nominated by expert bodies and the Federal government. All appeals need to be settled in greatest a few hearings or a interval of 6 months from the day of the submitting of the charm.

After that the up coming stage is to enlist the providers of an agent. There are quite a number out there. The 1 you pick should to be excellent enough for you in generating the decision. It has to be an sincere and reputable 1. A single could be chosen by acquiring approval from one of your family or colleagues. Appear at conditions and circumstances of functioning with different brokers then select the a single that you will be most comfortable with. Make certain that you sign contracts before you commence operating with them.

Yes, if you're reading this proper now, this is 1 of my on the web approaches. Running a blog is an wonderful resource in which you can share your feelings, techniques, suggestions, goods/services, and business. I have actually sponsored several people just through blogging.

This approach is accomplished with the assist of large machinery and there are individuals who are specialised in managing these kinds of equipment who do the work for us. Now you might feel that undertaking this would price you a bomb. You are proper, it would value you a great deal to comprehensive this entire approach, nonetheless, and you would not have to pay out something. That's right; you will not have to set a solitary penny out of your pocket due to the fact most of such assignments are funded by the federal government. The federal government has allocated massive sums of income to elevating homes that are in the flooding degree. Consequently all you are necessary to do is fill out some paper work and shortly your residence will be lifted out of the line of flooding and out of harms get to.

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