#author("2024-02-27T11:55:03+09:00","default:Miyashita","Miyashita") #author("2024-02-27T12:08:20+09:00","default:Miyashita","Miyashita") *デルタ関数 メモ [#u721e50b] #contents **性質 [#u51f4d25] よく \(\delta(x)\) とか \(\delta(t)\) のように書かれる．~ 定義より先に性質を見た方が感覚をつかみやすい． \[ \int_{-\infty}^{\infty} f(x)\delta(x-a) dx = f(a) \] 関数 \(f(x)\) のある特定の場所 \(x=a\) の値を抜き出す役割．~ \(\delta(x)\) の入力に対する応答が求められるのならば，畳み込み積分によって任意の入力 \( f(x) \)に対する応答が求まる．~ ということでグリーン関数でデルタ関数が出てくる． **定義？ [#hbdda6ca] プロットすることはできないが，\(\delta(x)\) は~ \(x=0\) まわりで無限に細く，無限に細いにもかかわらず面積は 1 となる何か，~ と思っておけば良い？ \[\begin{align} \delta(x) = 0, x\neq0 \\ \int_{-\varepsilon}^{\varepsilon} \delta(x) dx = 1 \end{align}\] **デルタ関数のいろいろな表現 [#ga2d3830] ***例1 [#p1753f7e] 式からはよくわからないが図を見ると確かにデルタ関数に近づいているような気がする． \[ \lim_{a \to 0} \frac{1}{\pi}\frac{a^2}{x^2+a^2} \] #htmlinsert(svg/delta1.svg) ***例2 [#a9577523] 正規分布の分散が0に近づいて，無限に細くなったような感じ．~ 個人的には一番わかりやすい． \[ \lim_{a \to 0} \frac{1}{a\sqrt{\pi}}\,\exp\,\left(-\frac{x^2}{a^2} \right) \] #htmlinsert(svg/delta2.svg) ***例3 [#m47ecb30] 振動数がどんどん激しくなるし，本当に \( n\to\infty\) でデルタ関数になる？とも感じる．~ 無限はすごい． \[ \lim_{n\to\infty} \frac{1}{\pi}\,\left( \frac{1}{2} + \sum_{k=1}^{n} \cos\,kx \right) \] #htmlinsert(svg/delta3.svg) ***例4 [#m498c110] 上に書いた式の親戚． \[ \lim_{n\to\infty} \frac{\sin\,nx}{\pi x} \] #htmlinsert(svg/delta4.svg) **例1-4を並べると [#m81c38e9] #htmlinsert(svg/delta1-4.svg) **参考文献 [#a6d4835b] //#htmlinsert(book_appliedmath_for_physics.html) -[[物理のための応用数学 小野寺嘉孝著 裳華房>https://www.amazon.co.jp/%25E7%2589%25A9%25E7%2590%2586%25E3%2581%25AE%25E3%2581%259F%25E3%2582%2581%25E3%2581%25AE%25E5%25BF%259C%25E7%2594%25A8%25E6%2595%25B0%25E5%25AD%25A6-%25E5%25B0%258F%25E9%2587%258E%25E5%25AF%25BA-%25E5%2598%2589%25E5%25AD%259D/dp/4785320311/ref=sr_1_2?__mk_ja_JP=%25E3%2582%25AB%25E3%2582%25BF%25E3%2582%25AB%25E3%2583%258A&crid=IMWSQNWFOT35&dib=eyJ2IjoiMSJ9.Y2JrQe3TjQrYE6piKndiNHxYRXxQ0oo8QZU0yt9p3CseiGneqnAG8KG0cp73PXGh4r7ommK1m3N3faXMV5jsPDuKzSLdA6-wnvONFAFBfeGBATgsIfZBpg5GaBRjpb3PPy_RgvjaDnzHI5ffDBVQWwUb6kYwKsEaw2kjii2LHc7FIDJOwe_Elis8b8s70WIANOvDDQgqYjwv9ArvtcOtDBCkTtkVDYaItXFnQzYYCWE.yZPvjiBlQF74Adizuumvhv2CzuOOvUG6JQIT4TXEC7A&dib_tag=se&keywords=%25E7%2589%25A9%25E7%2590%2586+%25E5%25B7%25A5%25E5%25AD%25A6%25E3%2581%25AE%25E3%2581%259F%25E3%2582%2581%25E3%2581%25AE+%25E5%25BF%259C%25E7%2594%25A8%25E6%2595%25B0%25E5%25AD%25A6&qid=1709002194&s=books&sprefix=%25E7%2589%25A9%25E7%2590%2586+%25E5%25B7%25A5%25E5%25AD%25A6%25E3%2581%25AE%25E3%2581%259F%25E3%2582%2581%25E3%2581%25AE+%25E5%25BF%259C%25E7%2594%25A8%25E6%2595%25B0%25E5%25AD%25A6%252Cstripbooks%252C386&sr=1-2&_encoding=UTF8&tag=hydrocoast-22&linkCode=ur2&linkId=cffde396a22a806dea5d9a01dd1e6a26&camp=247&creative=1211]] -[[物理のための応用数学 小野寺嘉孝(裳華房）>https://www.amazon.co.jp/%25E7%2589%25A9%25E7%2590%2586%25E3%2581%25AE%25E3%2581%259F%25E3%2582%2581%25E3%2581%25AE%25E5%25BF%259C%25E7%2594%25A8%25E6%2595%25B0%25E5%25AD%25A6-%25E5%25B0%258F%25E9%2587%258E%25E5%25AF%25BA-%25E5%2598%2589%25E5%25AD%259D/dp/4785320311/ref=sr_1_2?__mk_ja_JP=%25E3%2582%25AB%25E3%2582%25BF%25E3%2582%25AB%25E3%2583%258A&crid=IMWSQNWFOT35&dib=eyJ2IjoiMSJ9.Y2JrQe3TjQrYE6piKndiNHxYRXxQ0oo8QZU0yt9p3CseiGneqnAG8KG0cp73PXGh4r7ommK1m3N3faXMV5jsPDuKzSLdA6-wnvONFAFBfeGBATgsIfZBpg5GaBRjpb3PPy_RgvjaDnzHI5ffDBVQWwUb6kYwKsEaw2kjii2LHc7FIDJOwe_Elis8b8s70WIANOvDDQgqYjwv9ArvtcOtDBCkTtkVDYaItXFnQzYYCWE.yZPvjiBlQF74Adizuumvhv2CzuOOvUG6JQIT4TXEC7A&dib_tag=se&keywords=%25E7%2589%25A9%25E7%2590%2586+%25E5%25B7%25A5%25E5%25AD%25A6%25E3%2581%25AE%25E3%2581%259F%25E3%2582%2581%25E3%2581%25AE+%25E5%25BF%259C%25E7%2594%25A8%25E6%2595%25B0%25E5%25AD%25A6&qid=1709002194&s=books&sprefix=%25E7%2589%25A9%25E7%2590%2586+%25E5%25B7%25A5%25E5%25AD%25A6%25E3%2581%25AE%25E3%2581%259F%25E3%2582%2581%25E3%2581%25AE+%25E5%25BF%259C%25E7%2594%25A8%25E6%2595%25B0%25E5%25AD%25A6%252Cstripbooks%252C386&sr=1-2&_encoding=UTF8&tag=hydrocoast-22&linkCode=ur2&linkId=cffde396a22a806dea5d9a01dd1e6a26&camp=247&creative=1211]]