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In the field of [[mathematical analysis]], a '''general Dirichlet series''' is an [[series (mathematics)|infinite series]] that takes the form of


: <math>\sum_{n=1}^{\infty}a_n e^{-\lambda_n s},</math>


where <math>a_n</math>, <math>s</math> are [[complex number]]s and <math>\{\lambda_n\}</math> is a strictly increasing [[sequence]] of positive numbers that tends to infinity.
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A simple observation shows that an 'ordinary' [[Dirichlet series]]
 
: <math>\sum_{n=1}^{\infty}\frac{a_n}{n^s},</math>
 
is obtained by substituting <math>\lambda_n=\log n</math> while a [[power series]]
 
: <math>\sum_{n=1}^{\infty}a_n (e^{-s})^n,</math>
 
is obtained when <math>\lambda_n=n</math>.
 
== Fundamental theorems ==
 
If a Dirichlet series is convergent at <math>s_0=\sigma_0+t_0i</math>, then it is [[uniform convergence|uniformly convergent]] in the [[Domain of a function|domain]]
 
: <math>|\text{arg}(s-s_0)|\leq\theta<\frac{\pi}{2},</math>
 
and [[Convergent series|convergent]] for any <math>s=\sigma+ti</math> where <math>\sigma>\sigma_0</math>.
 
There are now three possibilities regarding the convergence of a Dirichlet series, i.e. it may converge for all, for none or for some values of ''s''. In the latter case, there exist a <math>\sigma_c</math> such that the series is convergent for <math>\sigma>\sigma_c</math> and [[divergent series|divergent]] for <math>\sigma<\sigma_c</math>. By convention, <math>\sigma_c=\infty</math> if the series converges nowhere and <math>\sigma_c=-\infty</math> if the series converges everywhere on the [[complex plane]].
 
== Abscissa of convergence ==
 
The '''abscissa of convergence''' of a Dirichlet series can be defined as <math>\sigma_c</math> above. Another equivalent definition is
 
: <math>\sigma_c=\inf\{\sigma\in\mathbb{R}:\sum_{n=1}^{\infty}a_n e^{-\lambda_n s} \text{ converges for any } s \text{ where Re}(s)>\sigma\}</math>.
 
The line <math>\sigma=\sigma_c</math> is called the '''line of convergence'''. The '''half-plane of convergence''' is defined as
 
: <math>\mathbb{C}_{\sigma_c}=\{s\in\mathbb{C}: \text{Re}(s)>\sigma_c\}.</math>
 
The [[abscissa]], [[line (geometry)|line]] and [[Half-space (geometry)|half-plane]] of convergence of a Dirichlet series are analogous to [[radius]], [[Boundary (topology)|boundary]] and [[disk (mathematics)|disk]] of convergence of a [[power series]].
 
On the line of convergence, the question of convergence remains open as in the case of power series. However, if a Dirichlet series converges and diverges at different points on the same vertical line, then this line must be the line of convergence. The proof is implicit in the definition of abscissa of convergence. An example would be the series
 
: <math>\sum_{n=1}^{\infty}\frac{1}{n}e^{-ns},</math>
 
which converges at <math>s=-\pi i</math> ([[harmonic series (mathematics)|alternating harmonic series]]) and diverges at <math>s=0</math> ([[harmonic series (mathematics)|harmonic series]]). Thus, <math>\sigma=0</math> is the line of convergence.
 
Suppose that a Dirichlet series does not converge at <math>s=0</math>, then it is clear that <math>\sigma_c\geq0</math> and <math>\sum a_n</math> diverges. On the other hand, if a Dirichlet series converges at <math>s=0</math>, then <math>\sigma_c\leq0</math> and <math>\sum a_n</math> converges. Thus, there are two formulas to compute <math>\sigma_c</math>, depending on the convergence of <math>\sum a_n</math> which can be determined by various [[convergence tests]]. These formulas are similar to the [[Cauchy–Hadamard theorem|Cauchy-Hadamard theorem]] for the radius of convergence of a power series.
 
If <math>\sum a_k</math> is divergent, i.e. <math>\sigma_c\geq0</math>, then <math>\sigma_c</math> is given by
 
: <math>\sigma_c=\limsup_{n\to\infty}\frac{\log|a_1+a_2+\cdots+a_n|}{\lambda_n}.</math>
 
If <math>\sum a_k</math> is convergent, i.e. <math>\sigma_c\leq0</math>, then <math>\sigma_c</math> is given by
 
: <math>\sigma_c=\limsup_{n\to\infty}\frac{\log|a_{n+1}+a_{n+2}+\cdots|}{\lambda_n}.</math>
 
== Abscissa of absolute convergence ==
 
A Dirichlet series is [[absolute convergence|absolutely convergent]] if the series
 
: <math>\sum_{n=1}^{\infty}|a_n e^{-\lambda_n s}|,</math>
 
is convergent. As usual, an absolutely convergent Dirichlet series is convergent, but the [[Theorem#Converse|converse]] is not always true.
 
If a Dirichlet series is absolutely convergent at <math>s_0</math>, then it is absolutely convergent for all ''s'' where <math>\text{Re}(s)>\text{Re}(s_0)</math>. A Dirichlet series may converge absolutely for all, for no or for some values of ''s''. In the latter case, there exist a <math>\sigma_a</math> such that the series converges absolutely for <math>\sigma>\sigma_a</math> and converges non-absolutely for <math>\sigma<\sigma_a</math>.
 
The '''abscissa of absolute convergence''' can be defined as <math>\sigma_a</math> above, or equivalently as
 
: <math>\sigma_c=\inf\{\sigma\in\mathbb{R}:\sum_{n=1}^{\infty}a_n e^{-\lambda_n s} \text{ converges absolutely for any } s \text{ where Re}(s)>\sigma\}</math>.
 
The '''line''' and '''half-plane of absolute convergence''' can be defined similarly. There are also two formulas to compute <math>\sigma_a</math>.
 
If <math>\sum |a_k|</math> is divergent, then <math>\sigma_a</math> is given by
 
: <math>\sigma_a=\limsup_{n\to\infty}\frac{\log(|a_1|+|a_2|+\cdots+|a_n|)}{\lambda_n}.</math>
 
If <math>\sum |a_k|</math> is convergent, then <math>\sigma_a</math> is given by
 
: <math>\sigma_a=\limsup_{n\to\infty}\frac{\log(|a_{n+1}|+|a_{n+2}|+\cdots)}{\lambda_n}.</math>
 
In general, the abscissa of convergence does not coincide with abscissa of absolute convergence. Thus, there might be a strip between the line of convergence and absolute convergence where a Dirichlet series is [[conditional convergence|conditionally convergent]]. The width of this strip is given by
 
: <math>0\leq\sigma_a-\sigma_c\leq L:=\limsup_{n\to\infty}\frac{\log n}{\lambda_n}.</math>
 
In the case where ''L''= 0, then
 
: <math>\sigma_c=\sigma_a=\limsup_{n\to\infty}\frac{\log |a_n|}{\lambda_n}.</math>
 
All the formulas provided so far still hold true for 'ordinary' [[Dirichlet series]] by substituting <math>\lambda_n=\log n</math>.
 
== Analytic functions ==
 
A [[Function (mathematics)|function]] represented by a Dirichlet series
 
: <math>f(s)=\sum_{n=1}^{\infty}a_n e^{-\lambda_n s},</math>
 
is [[Analytic function|analytic]] on the half-plane of convergence. Moreover, for <math>k=1,2,3,...</math>
 
: <math>f^{(k)}(s)=(-1)^k\sum_{n=1}^{\infty}a_n\lambda_n^k e^{-\lambda_n s}.</math>
 
== Further generalizations ==
 
A Dirichlet series can be further generalized to the [[variable (mathematics)|multi-variable]] case where <math>\lambda_n\in\mathbb{R}^k</math>, ''k'' = 2, 3, 4,..., or [[complex analysis|complex variable]] case where <math>\lambda_n\in\mathbb{C}^m</math>, ''m'' = 1, 2, 3,...
 
== References ==
 
* [[G. H. Hardy]], and M. Riesz, ''The general theory of Dirichlet's series'', Cambridge University Press, first edition, 1915.
* [[Edward Charles Titchmarsh|E. C. Titchmarsh]], ''The theory of functions'', Oxford University Press, second edition, 1939.
* [[Tom M. Apostol|Tom Apostol]], ''Modular functions and Dirichlet series in number theory'', Springer, second edition, 1990.
* A.F. Leont'ev, ''Entire functions and series of exponentials'' (in Russian), Nauka, first edition, 1982.
* A.I. Markushevich, ''Theory of functions of a complex variables'' (translated from Russian), Chelsea Publishing Company, second edition, 1977.
* [[J.-P. Serre]], ''A Course in Arithmetic'', Springer-Verlag, fifth edition, 1973.
 
== External links ==
 
* {{planetmath reference|title=Dirichlet series|id=4764}}
 
* {{Springer|title = Dirichlet series|id=d/d032920}}
 
[[Category:Complex analysis]]
[[Category:Mathematical series]]

Latest revision as of 02:26, 12 May 2014


The forex industry is a excellent place for individual investors, big and tiny, to engage in thrilling, fast-paced and possibly profitable trades. But you can not participate in forex currency trading if you do not very first have a forex brokerage account. Although most stock-marketplace brokerages enable you to also trade bonds, mutual funds, and other monetary instruments, forex brokerage accounts are typically standalone entities. Right here is what you want to know about opening a brokerage account.

Leverage

One particular of the major rewards of trading currencies is the tremendous quantity of leverage even small-time traders are permitted. Common leverage is 100:1, meaning for each $1 in your brokerage account, you can handle up to $one hundred in currencies. A thousand dollars would therefore permit you to handle $one hundred,000 worth of currency, so if the currency went up by 1% -- $1,000 -- you would truly double your cash! But if the currency went down by just 1%, you would lose all $1,000 of your investment. What would happen if the currency went down by 2%? Properly, theoretically, you would lose $1,000 above and beyond your initial investment, but in reality, a brokerage firm will typically step in and prevent this kind of loss.

Your principal decision is what level of leverage to apply for. Leverage is offered based on credit-worthiness, so if your credit report is fairly poor, you might want to pursue just 50:1 leverage -- which nevertheless gives you a lot of space to profit but limits your threat. Alternatively, if you have accurate nerves of steel and a actual knack for forex trading, you could be able to apply for as fantastic as 250:1 leverage!

Spreads

The excellent news is that there are no commissions charged on forex trades. The negative news is that, like stocks, forex currency pairs do have a bid/ask spread -- meaning a market place maker will pay much less for a currency than he is willing to sell it for. If you need to be taught further about cheap kids winter coats, there are many libraries you might pursue. These spreads are incredibly tiny, typically much less than .05 cents, but the wider the spread, the much more costly trading will be more than the extended run.

Not every brokerage has the exact same spreads, so it is crucial to overview the standard distance among the bid and ask costs just before choosing a broker.

Other Considerations

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