More Exercises

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1. Prove the chain rule formula: $ d(p_1(p_2(x)))=dp_1(p_2(x))\, dp_2(x)$.


2. Use calculus show that

$\displaystyle (e^{hd}\sin(x))\vert _{x\to 0}= h-\frac{h^3}{3!}+\frac{h^5}{5!}-\frac{h^7}{7!}+\dots,$     (6)


$\displaystyle (e^{hd}\cos(x))\vert _{x\to 0}= 1-\frac{h^2}{2!}+\frac{h^4}{4!}-\frac{h^6}{6!}+\dots\,.$     (7)


3. Denote $ \sin(h)$ and $ \cos(h)$ to be the right hand side power series in $ ($6$ )$ and $ ($7$ )$ respectively. Show that $ \sin^2(h)+\cos^2(h)=1$. Show that $ dsin(x)=cos(x)$, $ dcos(x)=-sin(x)$. Let $ i$ be a symbol, such that $ i^2=-1$. Show that $ e^{ih}=\cos(h)+i\sin(h)$.


4. Prove the formula

$\displaystyle \prod_{k=0}^{n-1}(1+xq^k)=\sum_{k=0}^n q^{\frac{k(k-1)}{2}}{n \choose k}_q x^k. $


5. Consider the area $ A$ under the graph of $ x^k$ on the segment $ [0,1]$. Let us chose $ q<1$ and divide $ [0,1]$ in infintely many segments $ [q^{n+1},q^n]$, $ n=0,1,2,\dots$ . Then our area $ A$ can be approximated by the sum $ s(q)$:

$\displaystyle A\simeq s(q):= \sum_{n=0}^\infty (q^n-q^{n+1})q^{nk}=\frac{1-q}{1-q^{k+1}}=\frac{1}{[k+1]_q}. $

Prove that $ A=\lim_{q\to 1} s(q)$.


6. Prove that

$\displaystyle \sum_{k=0}^n(-1)^k {n\choose k}_q=(1-q)(1-q^2)\dots (1-q^{n-1}), $

if $ n$ is even and 0 if $ n$ is odd.

7. Prove that $ D_qX-qXD_q=1$.