Maybe I am just being stupid, but I don't understand why in the Laplace inversion formula
(\mathcal{L}^{-1} F)(t) = \frac{1}{2\pi i} \int_{\sigma-i\infty}^{\sigma+i\infty} e^{st} F(s) ds
the contour of integration must be chosen so that \sigma is greater than the real part of all...
Denote by V(x) the speed of a particle at position x. Let's call v(x;\zeta) a measurement of it, which depends on some parameter \zeta, and denote the error by
\epsilon(x;\zeta)=v(x;\zeta)-V(x).
In order for the measurement to produce meaningful results, we must have some kind of error...
I have a quantity U(x), x>0, which I cannot calculate exactly. Numerically, I can calculate an approximation u(x;\eta), for \eta>0, which is very close to U(x) if \eta is small enough. I know that the error \xi(x;\eta)=u(x;\eta)-U(x) satisfies an estimate
|\xi(x;\eta)|\le E(x;\eta)
where...
A function f is both integrable and infinitely differentiable, i.e. f\in L_1(\mathbb{R}) \cap C^{\infty}(\mathbb{R}). Is it correct to say that this implies that the derivatives of f are also in L_1(\mathbb{R})? My reasoning: we have I<\infty, where
I=\int_{-\infty}^{\infty} f(x) dx = [x...
I am looking for a bound for the following expression
S=\sum_{n=1}^N n^k e^{-an}
where a>0 and k=1, 2, 3, or 4, apart from the obvious one:
S\le \frac{n+1}{2} \sum_{n=1}^N e^{-an} = \frac{n+1}{2}
\frac{1-e^{-Na}}{e^a-1}
Suppose I have two series
A=\sum_{n=0}^\infty a_n
B=\sum_{n=0}^\infty b_n
and I have estimates for the remainders of each one:
\sum_{n=N}^\infty a_n \le R^N_A
\sum_{n=N}^\infty b_n \le R^N_B
Consider the product series
AB=\sum_{n=0}^\infty c_n
where c_n=\sum_{i=0}^n a_i...
Is there a convenient shorthand for an "indicator function" of a positive integer n which vanishes if n is odd and is equal to 1 otherwise? I was thinking about something like
\mathbb{1}_{\{\mathrm{mod}(n,2)=0\}}(n),
but I'm not sure if this would be considered acceptable mathematical notation.
How do I calculate the integral
\int_{ix}^{i\infty} e^{-t} t^{-s-1}dt,
where x>0, s>0? Mathematica gives \Gamma(-s,ix), where \Gamma(\cdot,\cdot) is the incomplete gamma function, but I am not sure how to justify this formally.
Suppose we have the product
[(\pm ia) (\pm ib)]^{-\alpha}
wherea, b, \alpha >0. For which of the combinations (+,+), (+,-), (-,+), and (-,-) is the following property satisfied?
[(\pm ia) (\pm ib)]^{-\alpha}=(\pm ia)^{-\alpha} (\pm ib)^{-\alpha}