查看“︁半指數函數”︁的源代码

在[[數學]]上，'''半指數函數'''（Half-exponential function）是[[指數函數]]的{{link-en|函數平方根|Functional square root}}；換句話說，若<math>f</math>是一個半指數函數，則<math>f</math>與自己的[[複合函數]]會是一個指數函數：{{r|sqrtexp|miltersen}}
<math display=block>f\bigl(f(x)\bigr) = ab^x,</math>
其中{{nowrap|<math>a</math>與<math>b</math>.}}是常數。

==解析解的不存在性==
假若以加減乘除等標準算數運算、指數、對數及實數常數等來表達一個函數<math>f</math>，那麼<math>f\bigl(f(x)\bigr)</math>要不就是次指數的，要不就是超指數的，{{r|transseries}}因此{{link-en|哈代L-函數|Hardy field}}不可能是半指數函數。

==建構==
有無限多的函數，其半複合函數是與彼此相同的指數函數；特別地，對於任意位於[[開區間]]<math>(0,1)</math>當中的數<math>A</math>及任意從<math>[0,A]</math>映至<math>[A,1]</math>的[[單調函數|嚴格遞增]][[满射]][[連續函數|連續]]函数<math>g</math>而言，都存在作為這函數擴張的嚴格遞增連續實數函數<math>f</math>，使得{{nowrap|<math>f\bigl(f(x)\bigr)=\exp x</math>.{{r|croneu}}}}，而這<math>f</math>是以下[[函數方程]]的唯一解：

<math display=block> f (x) =
\begin{cases}
g (x) & \mbox{if } x \in [0,A], \\
\exp g^{-1} (x) & \mbox{if } x \in (A,1], \\
\exp f ( \ln x) & \mbox{if } x \in (1,\infty), \\
\ln f ( \exp x) & \mbox{if } x \in (-\infty,0). \\
\end{cases}
</math>

[[File:Half-exponential_function.png|thumb|right|300px|半指數函數的例子]]
一個簡單的、使得<math>f</math>處處有連續一階導數例子，是設<math>A=\tfrac12</math>且<math>g(x)=x+\tfrac12</math>，而這會得到下式： 
<math display=block> f (x) =
\begin{cases}
\log_e\left(e^x +\tfrac12\right) & \mbox{if } x \le -\log_e 2, \\
e^x - \tfrac12 & \mbox{if } -\log_e 2 \le x \le 0, \\
x +\tfrac12 & \mbox{if } 0 \le x \le \tfrac12, \\
e^{x-1/2} & \mbox{if } \tfrac12 \le x \le 1 , \\
x \sqrt{e} & \mbox{if } 1 \le x \le \sqrt{e} , \\
e^{x / \sqrt{e}} & \mbox{if } \sqrt{e} \le x \le e , \\
x^{\sqrt{e}} & \mbox{if } e \le x \le e^{\sqrt{e}} , \\
e^{x^{1/\sqrt{e}}}  & \mbox{if } e^{\sqrt{e}} \le x \le e^e , \ldots\\
\end{cases}
</math>

==應用==
半指數函數出現於[[計算複雜性理論]]當中，在其中半指數成長率是介於多項式成長率與指數成長率「之間」的一種成長速率。{{r|miltersen}}若一個函數<math>f</math>的成長率至少與半指數函數一樣快（也就是這函數與自身的複合函數的成長率是指數函數），就表示說這函數是[[單調函數|非遞減的]]，且對於任意<math>C>0</math>而言，有<math>f^{-1}(x^C)=o(\log x)</math>。{{r|razrud}}

==參見==
*[[迭代函數]]
*{{link-en|施羅德方程式|Schröder's equation}}
*{{link-en|阿貝爾方程式|Abel equation}}

==參考資料==
{{reflist|refs=

<ref name=croneu>{{cite journal
 | last1 = Crone | first1 = Lawrence J.
 | last2 = Neuendorffer | first2 = Arthur C.
 | doi = 10.1016/0022-247X(88)90080-7 | doi-access = free
 | issue = 2
 | journal = Journal of Mathematical Analysis and Applications
 | mr = 943525
 | pages = 520–529
 | title = Functional powers near a fixed point
 | volume = 132
 | year = 1988}}</ref>

<ref name=miltersen>{{cite conference
 | last1 = Miltersen | first1 = Peter Bro
 | last2 = Vinodchandran | first2 = N. V.
 | last3 = Watanabe | first3 = Osamu
 | editor1-last = Asano | editor1-first = Takao
 | editor2-last = Imai | editor2-first = Hiroshi
 | editor3-last = Lee | editor3-first = D. T.
 | editor4-last = Nakano | editor4-first = Shin{-}Ichi
 | editor5-last = Tokuyama | editor5-first = Takeshi
 | contribution = Super-polynomial versus half-exponential circuit size in the exponential hierarchy
 | doi = 10.1007/3-540-48686-0_21
 | mr = 1730337
 | pages = 210–220
 | publisher = Springer
 | series = Lecture Notes in Computer Science
 | title = Computing and Combinatorics, 5th Annual International Conference, COCOON '99, Tokyo, Japan, July 26-28, 1999, Proceedings
 | volume = 1627
 | year = 1999}}</ref>

<ref name=razrud>{{cite journal
 | last1 = Razborov | first1 = Alexander A. | author1-link = Alexander Razborov
 | last2 = Rudich | first2 = Steven | author2-link = Steven Rudich
 | doi = 10.1006/jcss.1997.1494 | doi-access = free
 | issue = 1
 | journal = [[Journal of Computer and System Sciences]]
 | mr = 1473047
 | pages = 24–35
 | title = Natural proofs
 | volume = 55
 | year = 1997}}</ref>

<ref name=sqrtexp>{{cite journal
 | last = Kneser
 | first = H.
 | author-link = Hellmuth Kneser
 | journal = [[Crelle's Journal|Journal für die reine und angewandte Mathematik]]
 | mr = 0035385
 | pages = 56–67
 | title = Reelle analytische Lösungen der Gleichung {{math|''φ''(''φ''(''x'') {{=}} ''e''<sup>''x''</sup>}} und verwandter Funktionalgleichungen
 | url = https://gdz.sub.uni-goettingen.de/id/PPN243919689_0187?tify%3D%7B%22pages%22%3A%5B62%5D%7D
 | volume = 187
 | year = 1950
 | access-date = 2022-06-22
 | archive-date = 2022-05-18
 | archive-url = https://web.archive.org/web/20220518013617/https://gdz.sub.uni-goettingen.de/id/PPN243919689_0187?tify%3D%7B%22pages%22:%5B62%5D%7D
 | dead-url = no
 }}</ref>

<ref name=transseries>{{cite book
 | last = van der Hoeven | first = J.
 | doi = 10.1007/3-540-35590-1
 | isbn = 978-3-540-35590-8
 | mr = 2262194
 | publisher = Springer-Verlag, Berlin
 | series = Lecture Notes in Mathematics
 | title = Transseries and real differential algebra
 | volume = 1888
 | year = 2006}}. See exercise 4.10, p. 91, according to which every such function has a comparable growth rate to an exponential or logarithmic function iterated an integer number of times, rather than the [[half-integer]] that would be required for a half-exponential function.</ref>

}}
==外部連結==
* [https://mathoverflow.net/q/12081 Does the exponential function have a (compositional) square root?]
* [https://mathoverflow.net/q/45477 “Closed-form” functions with half-exponential growth]

[[Category:算法分析]]
[[Category:計算複雜性理論]]