Difference between revisions of "Function of exponential type"
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+ | $#C+1 = 14 : ~/encyclopedia/old_files/data/F041/F.0401990 Function of exponential type | ||
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− | + | An [[Entire function|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 | then | ||
− | + | $$ | |
+ | \overline{\lim\limits}\; _ {k \rightarrow \infty } {| a _ {k} | } ^ {1/k} < \infty . | ||
+ | $$ | ||
− | The simplest examples of functions of exponential type are | + | 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 | 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 | + | where $ \gamma ( t) $ |
+ | is the function associated with $ f ( z) $ | ||
+ | in the sense of Borel (see [[Borel transform|Borel transform]]) and $ C $ | ||
+ | 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> | <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> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
− | |||
====References==== | ====References==== | ||
<table><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> | <table><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> |
Revision as of 19:40, 5 June 2020
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
$$ \overline{\lim\limits}\; _ {k \rightarrow \infty } {| a _ {k} | } ^ {1/k} < \infty . $$
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 |
Comments
References
[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=24449