# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

6 questions
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+50

### Evaluating sum $\sum_{m=0}^{\infty}\frac{(2-\delta_m^0)(-1)^m \lambda_0}{a(\lambda_0^2 -(\frac{m\pi}{a}))}\cos(m\pi x/a)$

How Can I evaluate the following sum\sum_{m=0}^{\infty}\frac{2-\delta_m^0}{a}\frac{(-1)^m \lambda_0}{\lambda_0^2 -(\frac{m\pi}{a})}\cos\left(\frac{m\pi x}{a}\right)=\frac{\cos(\lambda_0 x)}{\sin(\...
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+100

### Compute the derivative $\partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx$ in the sense of distributions

Let $u:\Omega\subset \mathbb R^N \to \mathbb R$ be bounded function that solves (in the sense of distributions) an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic ...
### Examples of increasing homeomorphism from $\mathbb{R}_+$ onto $\mathbb{R}_+$ satisfying some inequalities
Let $\varphi : \mathbb{R}_+ \to \mathbb{R}_+$ be an increasing homeomorphism satisfying $\varphi(0)=0,$ where $\mathbb{R}_+:=[0,\infty).$ For example, $\varphi(s)=\frac{s^3}{1+s^2}$ for \$s \in \...