# Tagged Questions

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### Show that $H_i=H_{n-i}$ and $\sum H_i=1$

We define $$H_i=\frac{1}{n}\frac{(-1)^{n-1}}{i!(n-1)!}\int_{0}^{n}\prod_{j=0,j\neq i}^{n}(x-j)dx$$ This is called the Newton-Cotes coefficient. Here is the exercise: First, convince yourself that ...
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From the wikipedia article about product integrals I can see that if our function is scalar, then to compute type I product integral we can just take exponential of a usual integral: $$\prod_a^b ... 1answer 56 views ### Do we need \mu, \nu to be \sigma-finite to show \int fg \ d(\mu \otimes \nu) = \int f \ d\mu \int g \ d\nu? The problem statement: Let (X, \mathcal F, \mu), (Y, \mathcal G, \nu) be \sigma-finite and f \in \mathcal L^1 (\mu), g \in \mathcal L^1 (\nu). Show that fg \in \mathcal L^1 (\mu \otimes ... 1answer 114 views ### Limit of an n-ary product Since a definite integral is defined as$$\lim_{n\to\infty} \sum_{i=0}^n f(x_i^*)\,\Delta x = \int_a^b f(x)\,dx$$and the integral is much easier to calcluate than a sum, if we change the sum to a ... 1answer 201 views ### Is there a “continuous product”? Is there a "continuous product" which is the limit of the discrete product \Pi, just like the integral \int is the limit of summation \sum. Thanks! 1answer 213 views ### Dyson series and T product (II) After reading the previous posts related to the Dyson series, I have decided to open a new thread because there is something that I am still not understanding. It concerns the expression:$$ ...
One of the most important tool in quantum mechanics is the Dyson series because it is the basis of the perturbative theory. There is a step in the derivation that I can't understand. $\{H(t_i)\}$ are ...
The arithmetic mean of $y_i ... y_n$ is: $$\frac{1}{n}\sum_{i=1}^n~y_i$$ For a smooth function $f(x)$, we can find the arithmetic mean of $f(x)$ from $x_0$ to $x_1$ by taking $n$ samples and using ...