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Taylor's series

2021-08-09 20:44 作者:哼M(jìn)M 0人讀過 | 我要投稿

泰勒無窮級(jí)數(shù)與麥克勞倫無窮級(jí)數(shù)的關(guān)系: 泰勒公式存在一個(gè)a點(diǎn),當(dāng)這個(gè)小點(diǎn)等于0的時(shí)候,泰勒無窮級(jí)數(shù)就是麥克勞倫級(jí)數(shù)了.

Taylor series of the function ?f ?at a (or about a or centered at a).This case arises frequently enough that it is given the special name Maclaurin series.

2021.8.8

傅里葉級(jí)數(shù)(fourier's series)

如何理解傅里葉級(jí)數(shù)?

????1.回憶傅里葉的公式 $f(x)%3Da_%7B0%7D%20%2B%5Csum_%7Bn%3D1%7D%5EF%5Ba_%7Bn%7Dcos(%5Cfrac%7Bn%5Cpi%20%7D%7Bl%7D%20x)%2B%20%20b_%7Bn%7Dsin(%5Cfrac%7Bn%5Cpi%20%7D%7Bl%7D)%20%5D%2C$ 那個(gè)F 表示無窮.

2.那公式里還有a0,an 和 bn啊! $a_%7Bo%7D%20%3D%20%5Cfrac%7B1%7D%7B2l%7D%20%5Cint_%7B-l%7D%5E%7Bl%7D%20f(x)dx%2C$

???????????????????????????????????????????? $a_%7Bn%7D%3D%5Cfrac%7B1%7D%7Bl%7D%20%5Cint_%7B-l%7D%5E%7Bl%7D%20f(x)cos(%5Cfrac%7Bn%5Cpi%20%7D%7Bl%7Dx%20)dx%2C$

???????????????????????????????????????????? $b_%7Bn%7D%20%3D%5Cfrac%7B1%7D%7Bl%7D%20%5Cint_%7B-l%7D%5E%7Bl%7D%20f(x)sin(%5Cfrac%7Bn%5Cpi%20%7D%7Bl%7Dx%20)dx$

???????????????????????????????????????????? $4%3Af(-x)%3D%5Cvert%20_%7Bf(z)%2Ceven%7D%5E%7Bf(-x)%2Codd%7D%20$

3.聯(lián)系下sin和cos的積分性質(zhì),在對(duì)稱的區(qū)間內(nèi)積分,呵呵就有偶函數(shù)(even)是Double,奇函數(shù)(odd)是0,發(fā)現(xiàn)了一個(gè)問題: $f(x)$ 與 $%5Csin%20(x)%20%E5%92%8C%5Ccos%20(x)%20$ 的乘積is odd or even, we should be certainly make sure. I have the conclusion :odd *even = odd, other is even.

4.那么,接下來就用到奇偶測(cè)試了,公式在上面.利用奇偶測(cè)試,確定在傅里葉級(jí)數(shù)中存在sin還是cos,當(dāng)然還有 $l$ 就是你選取的范圍,題目里一般叫周期(一般有 $%5Cpi%20$ ), function expansion is difficult .