Difference between revisions of "Function of exponential type"
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− | An [[ | + | An [[entire function]] $f(z)$ satisfying the condition |
− | satisfying the condition | ||
$$ | $$ | ||
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where $ \gamma ( t) $ | where $ \gamma ( t) $ | ||
is the function associated with $ f ( z) $ | is the function associated with $ f ( z) $ | ||
− | in the sense of Borel (see [[ | + | in the sense of Borel (see [[Borel transform]]) and $ C $ |
is a closed contour enclosing all the singularities of $ \gamma ( t) $. | is a closed contour enclosing all the singularities of $ \gamma ( t) $. | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1964) (Translated from Russian) {{MR|0156975}} {{ZBL|0152.06703}} </TD></TR> | + | <table> |
− | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1964) (Translated from Russian) {{MR|0156975}} {{ZBL|0152.06703}} </TD></TR> | |
− | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> R.P. Boas, "Entire functions" , Acad. Press (1954) {{MR|0068627}} {{ZBL|0058.30201}} </TD></TR> | |
− | + | </table> | |
− | |||
− |
Latest revision as of 20:18, 6 November 2023
An entire function $f(z)$ satisfying the condition
$$ | f ( z) | < A e ^ {a | z| } ,\ \ | z | < \infty ,\ A , a < \infty . $$
If $ f ( z) $ is represented by a series
$$ f ( z) = \sum _ { k=0 } ^ \infty \frac{a _ {k} }{k!} z ^ {k} , $$ then $$ \limsup _ {k \rightarrow \infty } {| a _ {k} | } ^ {1/k} < \infty . $$
The simplest examples of functions of exponential type are $ e ^ {cx} $, $ \sin \alpha z $, $ \cos \beta z $, and $ \sum _ {k=1} ^ {n} A _ {k} e ^ {a _ {k} z } $.
A function of exponential type has an integral representation
$$ f ( z) = \frac{1}{2 \pi i } \int\limits _ { C } \gamma ( t) e ^ {zt} d t , $$
where $ \gamma ( t) $ is the function associated with $ f ( z) $ in the sense of Borel (see Borel transform) and $ C $ is a closed contour enclosing all the singularities of $ \gamma ( t) $.
References
[1] | B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1964) (Translated from Russian) MR0156975 Zbl 0152.06703 |
[a1] | R.P. Boas, "Entire functions" , Acad. Press (1954) MR0068627 Zbl 0058.30201 |
Function of exponential type. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Function_of_exponential_type&oldid=51260