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<title>Half-exponential function</title>

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Half-exponential function

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In <a href="Mathematics" title="Mathematics">mathematics</a>, a half-exponential function is a <a href="Functional_square_root" title="Functional square root">functional square root</a> of an <a href="Exponential_function" title="Exponential function">exponential function</a>. That is, a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> <a href="Function_composition" title="Function composition">composed</a> with itself results in an exponential function:<a href="#cite_note-sqrtexp-1">[1]</a><a href="#cite_note-miltersen-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle f{\bigl (}f(x){\bigr )}=ab^{x},}</annotation>

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for some constants <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">.

<a href="Hellmuth_Kneser" title="Hellmuth Kneser">Hellmuth Kneser</a> first proposed a <a href="Holomorphic_function" title="Holomorphic function">holomorphic</a> construction of the solution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f{\bigl (}f(x){\bigr )}=e^{x}}">

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<annotation encoding="application/x-tex">{\displaystyle f{\bigl (}f(x){\bigr )}=e^{x}}</annotation>

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</math><img src="./3441ba675f40ccc1e8d320db9e1d06175726f8fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.18ex; height:3.176ex;" alt="{\displaystyle f{\bigl (}f(x){\bigr )}=e^{x}}" loading="lazy"> in 1950. It is closely related to the problem of extending <a href="Tetration" title="Tetration">tetration</a> to non-integer values; the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}^{\frac {1}{2}}a}">

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<annotation encoding="application/x-tex">{\displaystyle {}^{\frac {1}{2}}a}</annotation>

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</math><img src="./c767106e50c28f8bd6f88db8f435bbd46c91b363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.966ex; height:3.509ex;" alt="{\displaystyle {}^{\frac {1}{2}}a}" loading="lazy"> can be understood as the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f{\bigl (}1)}">

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<annotation encoding="application/x-tex">{\displaystyle f{\bigl (}1)}</annotation>

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</math><img src="./dc8553474f2114489d21f64fdae765ca782c52d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.411ex; height:3.176ex;" alt="{\displaystyle f{\bigl (}1)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f{\bigl (}x)}">

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<annotation encoding="application/x-tex">{\displaystyle f{\bigl (}x)}</annotation>

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</math><img src="./5b450f2af19b1ac891fca6e2df8d75a59ebfed8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.578ex; height:3.176ex;" alt="{\displaystyle f{\bigl (}x)}" loading="lazy"> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f{\bigl (}f(x){\bigr )}=a^{x}}">

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<annotation encoding="application/x-tex">{\displaystyle f{\bigl (}f(x){\bigr )}=a^{x}}</annotation>

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</math><img src="./23460f92e59b5ec659daa0048c2cde7b9932b7f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.327ex; height:3.176ex;" alt="{\displaystyle f{\bigl (}f(x){\bigr )}=a^{x}}" loading="lazy">. Example values from Kneser's solution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f{\bigl (}f(x){\bigr )}=e^{x}}">

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<annotation encoding="application/x-tex">{\displaystyle f{\bigl (}f(x){\bigr )}=e^{x}}</annotation>

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</math><img src="./3441ba675f40ccc1e8d320db9e1d06175726f8fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.18ex; height:3.176ex;" alt="{\displaystyle f{\bigl (}f(x){\bigr )}=e^{x}}" loading="lazy"> include <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f{\bigl (}0)\approx 0.49856}">

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<annotation encoding="application/x-tex">{\displaystyle f{\bigl (}0)\approx 0.49856}</annotation>

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</math><img src="./4294859aa43a91d01a08826a0f582f5d151ed00b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.131ex; height:3.176ex;" alt="{\displaystyle f{\bigl (}0)\approx 0.49856}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f{\bigl (}1)\approx 1.64635}">

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<annotation encoding="application/x-tex">{\displaystyle f{\bigl (}1)\approx 1.64635}</annotation>

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</math><img src="./1beb3524384a36a9fb617cfbecb6df04465e7fc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.131ex; height:3.176ex;" alt="{\displaystyle f{\bigl (}1)\approx 1.64635}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Impossibility_of_a_closed-form_formula">Impossibility of a closed-form formula</h2></div>

If a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is defined using the standard arithmetic operations, exponentials, <a href="Logarithm" title="Logarithm">logarithms</a>, and <a href="Real_number" title="Real number">real</a>-valued constants, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f{\bigl (}f(x){\bigr )}}">

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<annotation encoding="application/x-tex">{\displaystyle f{\bigl (}f(x){\bigr )}}</annotation>

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</math><img src="./72919274532d8c1e1a56014e8d8ca873d733621c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.826ex; height:3.176ex;" alt="{\displaystyle f{\bigl (}f(x){\bigr )}}" loading="lazy"> is either subexponential or superexponential.<a href="#cite_note-transseries-3">[3]</a> Thus, a <a href="Hardy_field#Examples" title="Hardy field">Hardy L-function</a> cannot be half-exponential.

<div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2></div>

Any exponential function can be written as the self-composition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(f(x))}">

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<annotation encoding="application/x-tex">{\displaystyle f(f(x))}</annotation>

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</math><img src="./f11f1c1b426c8f37ee0d2c5b9a60e413d8793913.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.505ex; height:2.843ex;" alt="{\displaystyle f(f(x))}" loading="lazy"> for infinitely many possible choices of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">. In particular, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> in the <a href="Open_interval" class="mw-redirect" title="Open interval">open interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}">

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<annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation>

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</math><img src="./c79c6838e423c1ed3c7ea532a56dc9f9dae8290b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}" loading="lazy"> and for every <a href="Continuous_function" title="Continuous function">continuous</a> <a href="Monotonic_function" title="Monotonic function">strictly increasing</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,A]}">

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<annotation encoding="application/x-tex">{\displaystyle [0,A]}</annotation>

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</math><img src="./0aa6ebe9fc37f3f23d497d0804e3d0b523ed87b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.233ex; height:2.843ex;" alt="{\displaystyle [0,A]}" loading="lazy"> <a href="Surjective_function" title="Surjective function">onto</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [A,1]}">

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<annotation encoding="application/x-tex">{\displaystyle [A,1]}</annotation>

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</math><img src="./1b0a31fb8e9bf831c956c520293e98f8bf383484.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.233ex; height:2.843ex;" alt="{\displaystyle [A,1]}" loading="lazy">, there is an extension of this function to a continuous strictly increasing function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> on the real numbers such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f{\bigl (}f(x){\bigr )}=\exp x}">

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<annotation encoding="application/x-tex">{\displaystyle f{\bigl (}f(x){\bigr )}=\exp x}</annotation>

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</math><img src="./9315c9eebfeaee7f557768e2ba097b753b248961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.194ex; height:3.176ex;" alt="{\displaystyle f{\bigl (}f(x){\bigr )}=\exp x}" loading="lazy">.<a href="#cite_note-croneu-4">[4]</a> The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is the unique solution to the <a href="Functional_equation" title="Functional equation">functional equation</a>

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<annotation encoding="application/x-tex">{\displaystyle 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}}}</annotation>

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A simple example, which leads to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> having a continuous first derivative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'}">

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<annotation encoding="application/x-tex">{\displaystyle f'}</annotation>

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</math><img src="./258eaada38956fb69b8cb1a2eef46bcb97d3126b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}" loading="lazy"> everywhere, and also causes

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</math><img src="./b0779ad8a77b626a01d90c58ffc6c69fc996b788.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.719ex; height:2.843ex;" alt="{\displaystyle f''\geq 0}" loading="lazy"> everywhere (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

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</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> is concave-up,

and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f'(x)}</annotation>

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</math><img src="./d0cd7d7c75340e779d82658e19d1720ce84ab127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.144ex; height:3.009ex;" alt="{\displaystyle f'(x)}" loading="lazy"> increasing,

for all real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">),

is to take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\tfrac {1}{2}}}">

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</math><img src="./d8962a8bd5d85bf953f3cf22dcb5b949257273dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.5ex; height:3.509ex;" alt="{\displaystyle A={\tfrac {1}{2}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=x+{\tfrac {1}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle g(x)=x+{\tfrac {1}{2}}}</annotation>

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</math><img src="./a236551fbd0ff1b7e6d9c9fe6cf03d9dfdcb3882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.182ex; height:3.509ex;" alt="{\displaystyle g(x)=x+{\tfrac {1}{2}}}" loading="lazy">, giving

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<annotation encoding="application/x-tex">{\displaystyle f(x)={\begin{cases}\log _{e}\left(e^{x}+{\tfrac {1}{2}}\right)&{\mbox{if }}x\leq -\log _{e}2,\\e^{x}-{\tfrac {1}{2}}&{\mbox{if }}{-\log _{e}2}\leq x\leq 0,\\x+{\tfrac {1}{2}}&{\mbox{if }}0\leq x\leq {\tfrac {1}{2}},\\e^{x-1/2}&{\mbox{if }}{\tfrac {1}{2}}\leq x\leq 1,\\x{\sqrt {e}}&{\mbox{if }}1\leq x\leq {\sqrt {e}},\\e^{x/{\sqrt {e}}}&{\mbox{if }}{\sqrt {e}}\leq x\leq e,\\x^{\sqrt {e}}&{\mbox{if }}e\leq x\leq e^{\sqrt {e}},\\e^{x^{1/{\sqrt {e}}}}&{\mbox{if }}e^{\sqrt {e}}\leq x\leq e^{e},\ldots \\\end{cases}}}</annotation>

</semantics>

</math>

Crone and Neuendorffer claim that there is no semi-exponential function f(x)

that is both (a) analytic and (b) always maps reals to reals.

The <a href="Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> solution above achieves goal (b) but not (a).

Achieving goal (a) is possible by writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{x}}</annotation>

</semantics>

</math><img src="./841c0d168e64191c45a45e54c7e447defd17ec6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.256ex; height:2.343ex;" alt="{\displaystyle e^{x}}" loading="lazy"> as a Taylor

series based at a fixpoint Q (there are an infinitude of such fixpoints,

but they all are nonreal complex,

for example <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=0.3181315+1.3372357i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q=0.3181315+1.3372357i}</annotation>

</semantics>

</math><img src="./e07664170c463ecf1847b5a21b9bf61641205c53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.473ex; height:2.509ex;" alt="{\displaystyle Q=0.3181315+1.3372357i}" loading="lazy">), making

Q also be a fixpoint of f, that is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(Q)=e^{Q}=Q}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(Q)=e^{Q}=Q}</annotation>

</semantics>

</math><img src="./5cf74bc3164b667f17b5fa3e63aee0ea1b23428e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.577ex; height:3.176ex;" alt="{\displaystyle f(Q)=e^{Q}=Q}" loading="lazy">,

then computing the <a href="Taylor_series" title="Taylor series">Maclaurin series</a> coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x-Q)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x-Q)}</annotation>

</semantics>

</math><img src="./b01c0f9fee39250f04d5a5fdf92122caf2f46950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.096ex; height:2.843ex;" alt="{\displaystyle f(x-Q)}" loading="lazy"> one by one. This results in Kneser's construction mentioned above.

<div class="mw-heading mw-heading2"><h2 id="Application">Application</h2></div>

Half-exponential functions are used in <a href="Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a> for growth rates "intermediate" between polynomial and exponential.<a href="#cite_note-miltersen-2">[2]</a> A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> grows at least as quickly as some half-exponential function (its composition with itself grows exponentially) if it is <a href="Monotonic_function" title="Monotonic function">non-decreasing</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(x^{C})=o(\log x)}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(x^{C})=o(\log x)}</annotation>

</semantics>

</math><img src="./e86bf2f216d7bc4018cb3e073ee204ed17f258cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.997ex; height:3.176ex;" alt="{\displaystyle f^{-1}(x^{C})=o(\log x)}" loading="lazy">, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C>0}</annotation>

</semantics>

</math><img src="./c84d4126c6df243734f9355927c026df6b0d3859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C>0}" loading="lazy">.<a href="#cite_note-razrud-5">[5]</a>

<ul><li><a href="Iterated_function" title="Iterated function">Iterated function</a> – Result of repeatedly applying a mathematical function</li>

<li><a href="Schr%C3%B6der's_equation" title="Schröder's equation">Schröder's equation</a> – Equation for fixed point of functional composition</li>

<li><a href="Abel_equation" title="Abel equation">Abel equation</a> – Equation for function that computes iterated values</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFKneser1950" class="citation journal cs1"><a href="Hellmuth_Kneser" title="Hellmuth Kneser">Kneser, H.</a> (1950). <a rel="nofollow" class="external text" href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0187?tify%3D%7B%22pages%22%3A%5B62%5D%7D">"Reelle analytische Lösungen der Gleichung φ(φ(x) = ex und verwandter Funktionalgleichungen"</a>. <a href="Crelle's_Journal" title="Crelle's Journal">Journal für die reine und angewandte Mathematik</a>. 187: 56–67. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1950.187.56">10.1515/crll.1950.187.56</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0035385">0035385</a>.</cite>

</li>

<li id="cite_note-miltersen-2">^ <a href="#cite_ref-miltersen_2-0">a</a> <a href="#cite_ref-miltersen_2-1">b</a> <cite id="CITEREFMiltersenVinodchandranWatanabe1999" class="citation conference cs1">Miltersen, Peter Bro; Vinodchandran, N. V.; Watanabe, Osamu (1999). "Super-polynomial versus half-exponential circuit size in the exponential hierarchy". In Asano, Takao; Imai, Hiroshi; Lee, D. T.; Nakano, Shin-ichi; Tokuyama, Takeshi (eds.). Computing and Combinatorics, 5th Annual International Conference, COCOON '99, Tokyo, Japan, July 26–28, 1999, Proceedings. Lecture Notes in Computer Science. Vol. 1627. Springer. pp. 210–220. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-48686-0_21">10.1007/3-540-48686-0_21</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-66200-6</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1730337">1730337</a>.</cite>

</li>

<li id="cite_note-transseries-3"><a href="#cite_ref-transseries_3-0">^</a> <cite id="CITEREFvan_der_Hoeven2006" class="citation book cs1">van der Hoeven, J. (2006). Transseries and Real Differential Algebra. Lecture Notes in Mathematics. Vol. 1888. Springer-Verlag, Berlin. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-35590-1">10.1007/3-540-35590-1</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-35590-8</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2262194">2262194</a>.</cite> 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 <a href="Half-integer" title="Half-integer">half-integer</a> that would be required for a half-exponential function.

</li>

<li id="cite_note-croneu-4"><a href="#cite_ref-croneu_4-0">^</a> <cite id="CITEREFCroneNeuendorffer1988" class="citation journal cs1">Crone, Lawrence J.; Neuendorffer, Arthur C. (1988). "Functional powers near a fixed point". Journal of Mathematical Analysis and Applications. 132 (2): 520–529. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-247X%2888%2990080-7">10.1016/0022-247X(88)90080-7</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0943525">0943525</a>.</cite>

</li>

<li id="cite_note-razrud-5"><a href="#cite_ref-razrud_5-0">^</a> <cite id="CITEREFRazborovRudich1997" class="citation journal cs1"><a href="Alexander_Razborov" title="Alexander Razborov">Razborov, Alexander A.</a>; <a href="Steven_Rudich" title="Steven Rudich">Rudich, Steven</a> (1997). <a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjcss.1997.1494">"Natural proofs"</a>. <a href="Journal_of_Computer_and_System_Sciences" title="Journal of Computer and System Sciences">Journal of Computer and System Sciences</a>. 55 (1): 24–35. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjcss.1997.1494">10.1006/jcss.1997.1494</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1473047">1473047</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/12081">Does the exponential function have a (compositional) square root?</a></li>

<li><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/45477">“Closed-form” functions with half-exponential growth</a></li></ul></div><div><div class="zim-footer">

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