Never regret. If it’s good, it’s wonderful. If it’s bad, it’s experience.
不必遗憾。若是美好，叫做精彩。若是糟糕，叫做经历。
设$a_1,a_2,cdots,a_ninmathbb{R}$.证明: $sum_{i,j=1}^nfrac{a_ia_j}{i+j}geq 0$.
注意到
egin{align*}
sum_{i,j=1}^nfrac{a_ia_j}{i+j}
&=sum_{i,j=1}^na_ia_jint_0^1x^{i+j-1}\,mathrm{d}x\
&=int_0^1frac{1}{x}sum_{i,j=1}^nleft(a_ix^i
ight)left(a_jx^j
ight)\,mathrm{d}x
=int_0^1frac{1}{x}left(sum_{i=1}^na_ix^i
ight)^2geq 0.
end{align*}
(hshhz)
(2020年北大数分)判断$f(x)=frac{x}{1+xcos^2 x}$在$[0,+infty)$上是否一致连续.
(2020年北大数分)设$q_kgeq p_k>0$, $q_{k+1}-q_kgeq p_k+p_{k+1}$且$sum_{k=1}^{infty}a_kln p_k=+infty$,记
egin{align*}
T_{p_k,q_k}(x) riangleq &frac{cos (q_k+p_k)x}{p_k}+frac{cos (q_k+p_k-1)x}{p_k-1}+frac{cos (q_k+p_k-2)x}{p_k-2}+cdots+frac{cos (q_k+1)x}{1}\
& -frac{cos (q_k-1)x}{1}-frac{cos (q_k-2)x}{2}-cdots-frac{cos (q_k-p_k)x}{p_k},
end{align*}
设$a_kgeq 0,sum_{k=1}^{infty}a_k<+infty$, $f(x)=sum_{k=1}^{infty}a_kT_{p_k,q_k}(x)$.
egin{enumerate}
item[(1)] 求证: $f(x)$是在$mathbb{R}$上连续的以$2pi$为周期的周期函数.
item[(2)] 判断并证明: $f(x)$的Fourier级数在$x=0$处的收敛性.
end{enumerate}
1. $f(x)$对任意$x_0in [a,b]$都上半连续,问$f(x)$在$[a,b]$上是否有最大值,给出证明或反例.
2. $f(x)$在$[1,+infty)$连续且满足:对任意$x,yin [1,+infty)$,有$f(x+y)leq f(x)+f(y)$.问$lim_{x o+infty}frac{f(x)}{x}$是否存在.
3. 已知$f(x)$在$[0,1]$连续,单调增加且$f(x)geq 0$,记
[s=frac{int_{0}^{1}xf(x)\,mathrm{d}x}{int_{0}^{1}f(x)\,mathrm{d}x}.]
egin{enumerate}[(1)]
item 证明$sgeq frac{1}{2}$.
item 比较$int_{0}^{s}f(x)\,mathrm{d}x$与$int_{s}^{1}f(x)\,mathrm{d}x$的大小. (可以用物理或几何直觉)
end{enumerate}
4.判断$f(x)=frac{x}{1+xcos^2 x}$在$[0,+infty)$上是否一致连续.
5.根据$int_{0}^{+infty}frac{sin x}{x}\,mathrm{d}x=frac{pi}{2}$,计算$int_{0}^{+infty}left(frac{sin x}{x} ight)^2\,mathrm{d}x$,并说明计算依据.
6.在承认平面Green公式的前提下证明如下特殊情况下的Stokes公式
[oint_Gamma R(x,y,z)\,mathrm{d}z=iint_Sigmafrac{partial R}{partial y}dydz-frac{partial R}{partial x}dzdx.]
7.设$0<p<1$,求$f(x)=cos px$在$[-pi,pi]$上的Fourier级数并求出其和函数,由此证明余元公式
[int_{0}^{1}x^{p-1}(1-x)^{-p}dx=frac{pi}{sin(ppi)}.]
8.设$C_r$为半径为$r$的圆周, $f(x,y)$满足$f(0,0)=0,frac{partial^2f}{partial x^2}+frac{partial^2f}{partial y^2}=x^2+y^2$, $f(x,y)$是$C^2$的,计算$A(r)=int_{C_r}f(x,y)\,mathrm{d}s$.
9.设$q_kgeq p_k>0$, $q_{k+1}-q_kgeq p_k+p_{k+1}$且$sum_{k=1}^{infty}a_kln p_k=+infty$,记
egin{align*}
T_{p_k,q_k}(x) riangleq &frac{cos (q_k+p_k)x}{p_k}+frac{cos (q_k+p_k-1)x}{p_k-1}+frac{cos (q_k+p_k-2)x}{p_k-2}+cdots+frac{cos (q_k+1)x}{1}\
& -frac{cos (q_k-1)x}{1}-frac{cos (q_k-2)x}{2}-cdots-frac{cos (q_k-p_k)x}{p_k},
end{align*}
设$a_kgeq 0,sum_{k=1}^{infty}a_k<+infty$, $f(x)=sum_{k=1}^{infty}a_kT_{p_k,q_k}(x)$.
egin{enumerate}
item[(1)] 求证: $f(x)$是在$mathbb{R}$上连续的以$2pi$为周期的周期函数.
item[(2)] 判断并证明: $f(x)$的Fourier级数在$x=0$处的收敛性.
end{enumerate}
1. $f(x)$对任意$x_0in [a,b]$都上半连续,问$f(x)$在$[a,b]$上是否有最大值,给出证明或反例.
2. $f(x)$在$[1,+infty)$连续且满足:对任意$x,yin [1,+infty)$,有$f(x+y)leq f(x)+f(y)$.问$displaystylelim_{x o+infty}frac{f(x)}{x}$是否存在.
3. 已知$f(x)$在$[0,1]$连续,单调增加且$f(x)geq 0$,记
$$s=frac{int_{0}^{1}xf(x)\,mathrm{d}x}{int_{0}^{1}f(x)\,mathrm{d}x}.$$
(1)证明$sgeq frac{1}{2}$.
(2)比较$displaystyleint_{0}^{s}f(x)\,mathrm{d}x$与$displaystyleint_{s}^{1}f(x)\,mathrm{d}x$的大小. (可以用物理或几何直觉)
4.证明$displaystyle f(x)=frac{xcos x}{1+sin^2x}$在$[0,+infty)$上一致连续.
5.根据$displaystyleint_{0}^{+infty}frac{sin x}{x}\,mathrm{d}x=frac{pi}{2}$,计算$displaystyleint_{0}^{+infty}left(frac{sin x}{x} ight)^2\,mathrm{d}x$,并说明计算依据.
6.在承认平面Green公式的前提下证明如下特殊情况下的Stokes公式
$$oint_Gamma R(x,y,z)\,mathrm{d}z=iint_Sigmafrac{partial R}{partial y}dydz-frac{partial R}{partial x}dzdx.$$
7.设$0< p<1$,求$f(x)=cos px$在$[-pi,pi]$上的Fourier级数,由此证明余元公式
$$int_{0}^{1}x^{p-1}(1-x)^{-p}dx=frac{pi}{sin(ppi)}.$$
8.设$C_r$为半径为$r$的圆周, $f(x,y)$满足$displaystyle f(0,0)=0,frac{partial^2f}{partial x^2}+frac{partial^2f}{partial y^2}=x^2+y^2$, $f(x,y)$是$C^2$的,计算$displaystyle A(r)=int_{C_r}f(x,y)\,mathrm{d}s$.
9.设$q_kgeq p_k>0$,
$$T_{p_k,q_k}(x)=frac{cos(p_k+1)x}{p_k}+cdots
+frac{cos(p_k+q_k)x}{p_k}-frac{cos(q_k+1)x}{q_k}-cdots
-frac{cos(q_k+p_k)x}{q_k}$$
(1) 证明$displaystyle f(x)=sum_{k=1}^{infty}a_kT_{p_k,q_k}(x)$是以$2pi$为周期的函数;
(2) $x=0$处收敛性. (注:题目不完整)