Everything about Entire Function totally explained
In
complex analysis, an
entire function, also called an
integral function, is a
function that's
holomorphic everywhere on the whole
complex plane. Typical examples of entire functions are the
polynomials, the
exponential function, and sums, products and compositions of these. Every entire function can be represented as a
power series which converges everywhere. Neither the
natural logarithm nor the
square root function is entire.
The
order of an entire function
is defined using the
limit superior as:
»
Note that an entire function may have a
singularity or even an
essential singularity at the complex
point at infinity. In the latter case, it's called a
transcendental entire function. As a consequence of
Liouville's theorem, a function which is entire on the whole
Riemann sphere (complex plane
and the point at infinity) is constant.
Liouville's theorem establishes an important property of entire functions — an entire function which is bounded must be constant. This property can be used for an elegant proof of the
fundamental theorem of algebra.
Picard's little theorem is a considerable strengthening of Liouville's theorem: a non-constant entire function takes on every complex number as value, except possibly one. The latter exception is illustrated by the
exponential function, which never takes on the value 0.
J. E. Littlewood chose the
Weierstrass sigma function as a 'typical' entire function in one of his books.
Further Information
Get more info on 'Entire Function'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://entire_function.totallyexplained.com">Entire function Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
