# 实分析作业代写Real analysis代考

## 极限 (数学)Limit (mathematics)代写

• 一致收敛Uniform convergence
• 紧空间Compact space
• 连续函数Continuous function
• 一致连续 Uniform continuity

## 实分析的相关

Various ideas from real analysis can be generalized from the real line to broader or more abstract contexts. These generalizations link real analysis to other disciplines and subdisciplines. For instance, generalization of ideas like continuous functions and compactness from real analysis to metric spaces and topological spaces connects real analysis to the field of general topology, while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the concepts of Banach spaces and Hilbert spaces and, more generally to functional analysis.

## 实分析课后作业代写

Using the logarithmic function defined in Section $7.1$, the reciprocal of any linear polynomial can be integrated. Indeed, up to a constant multiple, such a function is given by $1 /(x-\alpha)$, where $\alpha \in \mathbb{R}$, and we have
$$\frac{d}{d x}(\ln (x-\alpha))=\frac{1}{x-\alpha} \quad \text { for } x \in \mathbb{R}, x>\alpha$$
The next question that naturally arises is whether we can integrate the reciprocal of a quadratic polynomial, say $x^{2}+a x+b$, where $a, b \in \mathbb{R}$. If this quadratic happens to be the square of a linear polynomial, say $(x-\alpha)^{2}$, then the answer is easy because
$$\frac{d}{d x}\left(\frac{-1}{x-\alpha}\right)=\frac{1}{(x-\alpha)^{2}} \quad \text { for } x \in \mathbb{R}, x \neq \alpha .$$
Further, if the quadratic factors into distinct linear factors, that is, if $x^{2}+a x+b=(x-\alpha)(x-\beta) \quad$ for some $\alpha, \beta \in \mathbb{R}, \alpha>\beta$