Using the Taylor series expansion we have (for a sufficiently regular function $f$): $$ f(x+a)=\sum_{k=0}^n \frac{f^{(k)}(x)a^k}{k!} $$ So, defining the differential operator $D=\frac{d}{dx}$ and using the series expansion definition of the exponential function, we can write: $$ S_a f(x)=\exp(aD) f(x) $$ where $S_a f(x)=f(x+a)$ is the shift operator. This gives an ”intuitive” meaning to the […]

As the title says, I am looking for resources to learn some fixed point logic, especially partial fixed point logic. I have basic knowledge of propositional calculus and predicate logic, but sadly not much beyond that, and have need of PFP for my bachelor’s thesis. Thanks in advance.

For a math paper I need to be able to evaluate $\int_{-a}^{a}\delta^{(n)}(x)\ f(x)\ dx$ for differentiable $f$. I know that it is ‘supposed’ to equal $(-1)^nf^{(n)}(0)$: $$\int_{-a}^a\delta^{(n)}(x)\ f(x)\, dx=\int_{-a}^a \frac{d^n}{dx^n}\delta(x) f(x)\, dx =\int_{-a}^a(-1)^n\delta(x) \frac{d^nf}{dx^n}\, dx=(-1)^nf^{(n)}(0).$$ This argument seems like a hack and I have no idea what is actually going on. I want to write […]

I’m looking for a text that covers roughly what’s sometimes called “Calculus III” or multivariable calculus.* But this text must satisfy certain additional criteria: (1) It must be more in-depth (and consequently have harder exercises) than usual; (2) It must contain many solutions to the exercises; (3) It must not engage in the sort of […]

My gf is the classic math-phobe, totally traumatized by math, etc., which surprises me, since she’s whip-smart. The only explanation I can think of is that she got off to a bad start in elementary school. My own love for mathematics was kindled by an old book on number theory that I found in my […]

It should be well known that a Kleinian group and all its normal (non-elementary) subgroups have the same limit set. Do you know any book/article where I could find the proof? Thank you.

What more can be said about the identity derived from law of cosines (motivation below)$$ \cos(\widehat{A})BC+ A\cos(\widehat{B})C+ AB\cos(\widehat{C})+=\frac {A^2 + B^2 + C^2}{2} \tag{IV}$$ RHS seems as if operator $\cos(\widehat{\phantom{X}})$ is being applied consecutively to terms of ABC, I tried to represent it in an analogous way to the Laplacian operator convention, but maybe there […]

A friend of me who is not studying mathematics now needs to deal with integrals, double integrals and triple integrals within his study of chemistry. He asked me to give him a suggestion for a basic book that explains basic facts, rules etc., about integrals. I think this means Riemann integral and that it should […]

I need some references for starting to study about wavelet. I have enough information about abstract Harmonic analysis. Thanks!

Consider the Hilbert Space of weighted-square-integrable functions f(x): $$ \begin{equation} \int_{-\infty}^{\infty}\frac{f(x)^2}{e^{x}+e^{-x}}dx<\infty. \end{equation} $$ Note this integral is taken over the real line in both directions, and the weighting is roughly exponential. Hilbert Polynomials are complete using a different weighting, while Laguerre polynomials use a similar exponential weighting, but only on the positive real line. What […]

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Squared binomial coefficient
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Cubic with repeated roots has a linear factor
Finding the Limit in: $\lim\limits_{x\rightarrow1}\frac{\frac{1}{\sqrt{x}}-1}{x-1}$
Integrate $I(a) = \int_0^{\pi/2} \frac{dx}{1-a\sin x}$
Indefinite summation of polynomials