数学代写|MATH4331 Mathematical Analysis
Statistics-lab™可以为您提供uh.edu MATH4331 Mathematical Analysis数学分析的代写代考和辅导服务！
MATH4331 Mathematical Analysis课程简介
Precalculus is a foundational course that prepares students for more advanced math and science courses, such as calculus. The topics you listed are all important concepts and techniques that students need to understand in order to be successful in calculus and other advanced math and science courses.
Rational exponents involve expressing numbers with fractional exponents, which is a way of extending the concept of powers to include non-integer exponents. Circles involve studying the properties of circles and their equations. Functions are mathematical objects that represent relationships between inputs and outputs, and understanding their properties is essential for studying calculus. Complex numbers are numbers that include both real and imaginary components and are used extensively in advanced math and science courses.
PREREQUISITES
Synthetic division is a technique for dividing polynomials by linear factors. Inverse functions are functions that undo the effects of other functions, and they are important for understanding calculus. Exponential and logarithmic functions are functions that involve the constants e and log base 10 or log base e, respectively, and they are used extensively in science and engineering.
Overall, precalculus is an important course for students who are interested in pursuing advanced math and science courses. By mastering the concepts and techniques covered in this course, students will be well-prepared for the challenges of calculus and beyond.
MATH4331 Mathematical Analysis HELP（EXAM HELP， ONLINE TUTOR）
Lemma 2 Let $f$ be $2 \pi$-periodic, continuous on $\mathbb{R}$ and piecewise regular. Then
$$
k a_k=-b_k^{\prime}, \quad k b_k=a_k^{\prime}, \quad \forall k \in \mathbb{N},
$$
where the Fourier coefficients $a_k^{\prime}$ and $b_k^{\prime}$ of the derivative $f^{\prime}$ are given by
$$
a_k^{\prime}=\frac{1}{\pi} \int_{-\pi}^\pi f^{\prime}(x) \cos k x d x, \quad b_k^{\prime}=\frac{1}{\pi} \int_{-\pi}^\pi f^{\prime}(x) \sin k x d x .
$$
Proof As $f$ is continuous and piecewise regular, integrating by parts we obtain
$$
\begin{aligned}
k a_k & =\frac{k}{\pi} \int_{-\pi}^\pi f(x) \cos k x d x=\frac{1}{\pi}[f(x) \sin k x]{-\pi}^\pi-\frac{1}{\pi} \int{-\pi}^\pi f^{\prime}(x) \sin k x d x= \
& =-\frac{1}{\pi} \int_{-\pi}^\pi f^{\prime}(x) \sin k x d x=-b_k^{\prime} .
\end{aligned}
$$
The second relationship in (1.75) is similar.
Theorem 1 (Abel) Let $\sum_{k=0}^{\infty} a_k$ be a convergent numerical series and set
$$
f(x)=\sum_{k=0}^{\infty} a_k x^k, \quad-1<x<1 .
$$
Then as $x \rightarrow 1^{-}$the function $f(x)$ converges to $\sum_{k=0}^{\infty} a_k$.
Proof Take the sequence $s_n$ of partial sums of the given numerical series
$$
s_n=\sum_{k=0}^n a_k, \quad n=0,1,2, \ldots,
$$
and define $s_n$ also for $n=-1$ by $s_{-1}=0$. Given that $s_n-s_{n-1}=a_n$ for any $n=0,1,2, \ldots$, if $|x|<1$ we have
$$
\begin{aligned}
& \sum_{k=0}^n a_k x^k=\sum_{k=0}^n\left(s_k-s_{k-1}\right) x^k= \
& =s_n x^n+\sum_{k=0}^{n-1} s_k x^k-x \cdot \sum_{k=0}^n s_{k-1} x^{k-1}= \
& =s_n x^n+(1-x) \cdot \sum_{k=0}^{n-1} s_k x^k, \quad-1<x<1 . \
&
\end{aligned}
$$
Textbooks
• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.
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