復(fù)習(xí)筆記Day107：華中科技大學(xué)2023數(shù)學(xué)分析參考答案（下）

2023-02-23 23:57 作者:間宮_卓司 0人讀過 | 我要投稿

（續(xù)上）

7.設(shè) $f%3A%5Cleft%5B%200%2C1%20%5Cright%5D%20%5Crightarrow%20%5Cleft(%200%2C%2B%5Cinfty%20%5Cright)%20$ 為連續(xù)函數(shù)，常數(shù) $a%5Cge1$ ，證明

$%5Cunderset%7Bn%5Crightarrow%20%5Cinfty%7D%7B%5Clim%7D%5Csqrt%5Bn%5D%7B%5Cint_0%5E1%7B%5Cleft(%20a%2Bx%5En%20%5Cright)%20%5Enf%5Cleft(%20x%20%5Cright)%20%5Cmathrm%7Bd%7Dx%7D%7D%3Da%2B1$

這個(gè)感覺就是把典中典的題目魔改了一下，一方面，設(shè) $f%5Cleft(%20x%20%5Cright)%20%3CM%2CM%3E0$ ，則

$%5Cint_0%5E1%7B%5Cleft(%20a%2Bx%5En%20%5Cright)%20%5Enf%5Cleft(%20x%20%5Cright)%20%5Cmathrm%7Bd%7Dx%7D%3CM%5Cleft(%20a%2B1%20%5Cright)%20%5En$

另一方面，因?yàn)?img type="latex" class="latex" src="https://api.bilibili.com/x/web-frontend/mathjax/tex?formula=f(x)" alt="f(x)">連續(xù)，所以可以設(shè) $f(x)%3Em%3E0$ ，那么

$%5Cint_0%5E1%7B%5Cleft(%20a%2Bx%5En%20%5Cright)%20%5Enf%5Cleft(%20x%20%5Cright)%20%5Cmathrm%7Bd%7Dx%7D%3Em%5Cint_%7B1-%5Cfrac%7B2%7D%7Bn%5E2%7D%7D%5E%7B1-%5Cfrac%7B1%7D%7Bn%5E2%7D%7D%7B%5Cleft(%20a%2Bx%5En%20%5Cright)%20%5En%5Cmathrm%7Bd%7Dx%7D%3E%5Cfrac%7Bm%7D%7Bn%5E2%7D%5Cleft(%20a%2B%5Cleft(%201-%5Cfrac%7B2%7D%7Bn%5E2%7D%20%5Cright)%20%5En%20%5Cright)%20%5En$

兩邊同時(shí)開 $n$ 次方取極限可得結(jié)論

8.設(shè) $f(x)$ 在 $(-%5Cinfty.%2B%5Cinfty)$ 上可導(dǎo)，且對任意 $x$ 有 $f(x)%3Df(x%2B2k)%3Df(x%2Bb)$ ，其中 $k$ 為正整數(shù)， $b$ 為無理數(shù)，用 $%5Ctext%7BFourier%7D$ 級(jí)數(shù)理論證明 $f(x)$ 為常數(shù)

聽說是某年的競賽題，不過我沒有看答案，下面的方法對不對我也不清楚

設(shè) $f(x)$ 的最小正周期為 $2T$ ，若 $T%3D0$ ，那么依65.1，結(jié)論已經(jīng)成立了，現(xiàn)在設(shè) $T%3E0$ ，那么在 $%5B-T%2CT%5D$ 上

$f%5Cleft(%20x%20%5Cright)%5Csim%20%20%5Cfrac%7Ba_0%7D%7B2%7D%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%7B%5Cleft(%20a_n%5Ccos%20%5Cfrac%7Bn%5Cpi%7D%7BT%7Dx%2Bb_n%5Csin%20%5Cfrac%7Bn%5Cpi%7D%7BT%7Dx%20%5Cright)%7D$

又因?yàn)?img type="latex" class="latex" src="https://api.bilibili.com/x/web-frontend/mathjax/tex?formula=f(x)" alt="f(x)">在 $(-%5Cinfty.%2B%5Cinfty)$ 上可導(dǎo)，所以 $f(x)$ 滿足李普希茲條件，故 $f(x)$ 收斂于它的傅里葉級(jí)數(shù)，也就是

$f%5Cleft(%20x%20%5Cright)%20%3D%5Cfrac%7Ba_0%7D%7B2%7D%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%7B%5Cleft(%20a_n%5Ccos%20%5Cfrac%7Bn%5Cpi%7D%7BT%7Dx%2Bb_n%5Csin%20%5Cfrac%7Bn%5Cpi%7D%7BT%7Dx%20%5Cright)%7D%2C%5Cforall%20x%5Cin%20R$

記 $%5Cfrac%7B%5Cpi%7D%7BT%7D%3Dw$ ，因?yàn)?img type="latex" class="latex" src="https://api.bilibili.com/x/web-frontend/mathjax/tex?formula=%5Ccos%20%5Cfrac%7Bn%5Cpi%7D%7BT%7Dx%2C%5Csin%20%5Cfrac%7Bn%5Cpi%7D%7BT%7Dx" alt="%5Ccos%20%5Cfrac%7Bn%5Cpi%7D%7BT%7Dx%2C%5Csin%20%5Cfrac%7Bn%5Cpi%7D%7BT%7Dx">線性無關(guān)，所以

$%5Ccos%20wx%3D%5Ccos%20%5Cleft(%20w%5Cleft(%20x%2B2k%20%5Cright)%20%5Cright)%20%3D%5Ccos%20%5Cleft(%20w%5Cleft(%20x%2Bb%20%5Cright)%20%5Cright)%20$

這意味著 $%5Ccos%20wx$ 既以 $2k$ 為周期，又以 $b$ 為周期，而 $%5Ccos%20wx$ 以 $T%3D%5Cfrac%7B2%5Cpi%7D%7Bw%7D$ 為最小正周期，所以存在整數(shù) $N%2CM$ ，使得 $%5Cfrac%7B2%5Cpi%7D%7Bw%7D%3D2Nk%3DMb$ ，那么 $%5Cfrac%7BN%7D%7BM%7D%3D%5Cfrac%7Bb%7D%7B2k%7D$ ，但是左邊是有理數(shù)，右邊是無理數(shù)，矛盾

（這出現(xiàn)了偽證嗎？而且其中求傅里葉級(jí)數(shù)完全是多余的）

9.設(shè)二元函數(shù) $f(x%2Cy)$ 在 $(x_0%2Cy_0)$ 的某鄰域 $U$ 內(nèi)有定義，且在 $U$ 內(nèi)存在偏導(dǎo)數(shù)。證明：若

$f_x%5Cleft(%20x%2Cy%20%5Cright)%20%2Cf_y%5Cleft(%20x%2Cy%20%5Cright)%20$ 都在 $(x_0%2Cy_0)$ 可微，則 $f_%7Bxy%7D%5Cleft(%20x_0%2Cy_0%20%5Cright)%20%3Df_%7Byx%7D%5Cleft(%20x_0%2Cy_0%20%5Cright)%20$

這道題的證明思路類似于課本上證明若 $f_%7Bxy%7D$ 和 $f_%7Byx%7D$ 連續(xù)，則 $f_%7Bxy%7D%3Df_%7Byx%7D$

依題意，有

$%5Cbegin%7Baligned%7D%0A%09%26%5Cleft(%20f%5Cleft(%20x%2B%5CDelta%20x%2Cy%20%2B%5CDelta%20y%5Cright)%20-f%5Cleft(%20x%2B%5CDelta%20x%2Cy_0%20%5Cright)%20%5Cright)%20-%5Cleft(%20f%5Cleft(%20x_0%2Cy_0%2B%5CDelta%20y%20%5Cright)%20-f%5Cleft(%20x_0%2Cy_0%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3D%5CDelta%20x%5Cleft(%20f_x%5Cleft(%20x_0%2B%5Ctheta%20%5CDelta%20x%2Cy_0%2B%5CDelta%20y%20%5Cright)%20-f_x%5Cleft(%20x_0%2B%5Ctheta%20%5CDelta%20x%2Cy_0%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3D%5CDelta%20x%5Cleft(%20f_x%5Cleft(%20x_0%2Cy_0%20%5Cright)%20%2Bf_%7Bxx%7D%5Cleft(%20x_0%2Cy_0%20%5Cright)%20%5Ctheta%20%5CDelta%20x%2Bf_%7Bxy%7D%5Cleft(%20x_0%2Cy_0%20%5Cright)%20%5CDelta%20y%2B%20%5Cright.%5C%5C%0A%09%26%5Cleft.%20%2Bo%5Cleft(%20%5Csqrt%7B%5CDelta%20x%5E2%2B%5CDelta%20y%5E2%7D%20%5Cright)%20-f_x%5Cleft(%20x_0%2Cy_0%20%5Cright)%20%2Bf_%7Bxx%7D%5Cleft(%20x_0%2Cy_0%20%5Cright)%20%5Ctheta%20%5CDelta%20x%2Bo%5Cleft(%20%5CDelta%20x%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3D%5CDelta%20x%5CDelta%20yf_%7Bxy%7D%5Cleft(%20x_0%2Cy_0%20%5Cright)%20%2Bo%5Cleft(%20%5Csqrt%7B%5CDelta%20x%5E2%2B%5CDelta%20y%5E2%7D%20%5Cright)%5C%5C%0A%5Cend%7Baligned%7D$

同理 $%5Cbegin%7Baligned%7D%0A%09%26%5Cleft(%20f%5Cleft(%20x%2B%5CDelta%20x%2Cy%2B%5CDelta%20y%20%5Cright)%20-f%5Cleft(%20x%2Cy_0%2B%5CDelta%20y%20%5Cright)%20%5Cright)%20-%5Cleft(%20f%5Cleft(%20x_0%2B%5CDelta%20x%2Cy_0%20%5Cright)%20-f%5Cleft(%20x_0%2Cy_0%20%5Cright)%20%5Cright)%5C%5C%0A%09%26%3D%5CDelta%20x%5CDelta%20yf_%7Byx%7D%5Cleft(%20x_0%2Cy_0%20%5Cright)%20%2Bo%5Cleft(%20%5Csqrt%7B%5CDelta%20x%5E2%2B%5CDelta%20y%5E2%7D%20%5Cright)%5C%5C%0A%5Cend%7Baligned%7D$