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http:///mymemo.xyz/wiki/api.php?action=feedcontributions&user=221.248.215.176&feedformat=atom 2024-05-01T22:59:57Z 利用者の投稿記録 MediaWiki 1.31.0 2016-07-21T12:32:33Z

<p>221.248.215.176: 23.14069 26327 79269 00572 90863 67948 54738 02661 06242 60021 19934 45046 40952 43423 50690 45278 35169 71997 06754 92196 76というものを加えた</p> <hr /> <div>&#039;&#039;&#039;ゲルフォントの定数&#039;&#039;&#039;（ゲルフォントのていすう、{{lang-en|Gelfond&#039;s constant}}）は[[数学定数]]の一つで、[[ネイピア数]] &#039;&#039;e&#039;&#039; と[[円周率]] {{π}} を用いて &#039;&#039;e&#039;&#039;{{sup|{{π}}}} と表される数である。[[小数]]表示は<br /> :&#039;&#039;e&#039;&#039;{{sup|{{π}}}} = 23.14069 26327 79269 00572 90863 67948 54738 02661 06242 60021 19934 45046 40952 43423 50690 45278 35169 71997 06754 92196 76<br /> である。この数は[[ロシア]]の[[数学者]]{{仮リンク|アレクサンドル・ゲルフォント|en|Alexander Gelfond}}にちなんで名付けられた。ゲルフォントの定数は &#039;&#039;e&#039;&#039; や {{π}} と同様に[[超越数]]である。このことは[[ゲルフォント＝シュナイダーの定理]]から証明できる。<br /> <br /> == 数学的性質 ==<br /> &#039;&#039;e&#039;&#039;{{sup|{{π}}}} は[[オイラーの公式]]<br /> :&lt;math&gt;e^{ix} =\cos x+i\sin x&lt;/math&gt;<br /> から以下のように変形できる。ここで &#039;&#039;i&#039;&#039; は[[虚数単位]]である。<br /> :&lt;math&gt;e^{\pi} =(e^{i \pi} )^{-i} =(\cos \pi +i\sin \pi )^{-i} =(-1)^{-i}&lt;/math&gt;<br /> ゲルフォント＝シュナイダーの定理は 「&#039;&#039;a&#039;&#039; を 0, 1 でない[[代数的数]]、&#039;&#039;b&#039;&#039; を[[有理数]]でない代数的数とすると、&#039;&#039;a{{sup|b}}&#039;&#039; は超越数である」という内容である。&#039;&#039;a&#039;&#039; = &amp;minus;1, &#039;&#039;b&#039;&#039; = &amp;minus;&#039;&#039;i&#039;&#039; はこの条件を満たすので、(&amp;minus;1){{sup|&amp;minus;&#039;&#039;i&#039;&#039;}} は超越数である。すなわち &#039;&#039;e&#039;&#039;{{sup|{{π}}}} は超越数である。<br /> <br /> ちなみに、&#039;&#039;e{{sup|e}}&#039;&#039;, {{π}}{{sup|{{π}}}}, {{π}}{{sup|&#039;&#039;e&#039;&#039;}} などは[[有理数]]かどうか分かっていない。つまりこれらは超越数かどうかは知られていない。<br /> <br /> &lt;math&gt;k_0 =\frac{1}{\sqrt{2}}&lt;/math&gt; として<br /> :&lt;math&gt;k_n =\frac{1-\sqrt{1-{k_{n-1}}^2}}{1+\sqrt{1-{k_{n-1}}^2}}&lt;/math&gt;<br /> と定義されるとき、 [[数列]]<br /> :&lt;math&gt;\left( \frac{4}{k_{n+1}} \right)^{2^{-n}}&lt;/math&gt; <br /> は &#039;&#039;e&#039;&#039;{{sup|{{π}}}} &amp;minus; {{π}} に[[収束]]する。<br /> <br /> &#039;&#039;e&#039;&#039;{{sup|{{π}}}} &amp;minus; {{π}} は[[ほとんど整数]]である。<br /> :&#039;&#039;e&#039;&#039;{{sup|{{π}}}} &amp;minus; {{π}} = 19.99909997918947…<br /> <br /> == 関連項目 ==<br /> *[[ゲルフォント＝シュナイダーの定理]]<br /> *[[無理数]]<br /> *[[超越数]]<br /> <br /> == 外部リンク ==<br /> *{{MathWorld|urlname=GelfondsConstant|title=Gelfond&#039;s Constant}}<br /> <br /> {{DEFAULTSORT:けるふおんとのていすう}}<br /> [[Category:数学定数]]<br /> [[Category:超越数]]<br /> [[Category:数学に関する記事]]</div>

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