## integral form and specialization of a quantified algebra

Suppose I have an algebra $$A_K$$ set on $$K = mathbb {C} (x, y)$$. For example, let's say $$A_K$$ is generated by $$E, F$$ with relation $$(E, F) = frac {x ^ a-x ^ {a}} {y ^ b-y ^ {- b}} quad (*)$$
or $$a, b in mathbb {Z}$$, depend on $$E$$ and $$F$$. I'm trying to define a specialization of $$A_K$$ to an algebra $$A _ { mathbb {C}}$$ more than $$mathbb {C}$$, in which I force the relationship $$(E, F) = frac {a} {b}$$
hold, instead of $$(*)$$.

What I did, I considered algebra $$mathbb {C} (x ^ { pm}, y ^ { pm1}),$$ generated by $$E, F$$ with relation $$(*)$$. Then I specialize $$x = y$$, to get an algebra $$A_x$$ more than $$mathbb {C} (x ^ { pm1})$$ with relation $$(*)$$ to become $$(E, F) = frac {x ^ a-x ^ {a}} {x ^ b-x ^ {- b}}.$$
Now i get my algebra $$A_ mathbb {C}$$ by specializing $$x = 1$$.

My question is. Is this specialization mathematically correct? Is there a way to formalize it?

## computation – Find the main order contribution to a certain integral.

I am trying to calculate the main order term of the following expression in the little $$epsilon$$ limit;
$$I = frac {d} {ds} biggr | _ {s = 0} frac {1} { Gamma (s)} int_ {0} ^ { infty} dt frac {t ^ {s-1} e ^ {itx}} {(1-e ^ {i epsilon_ {1}}) (1-e ^ {i epsilon_ {2} t})}$$

First of all, I tried to extend the exponentials with $$epsilon$$is in them, leading to
$$– frac {1} { epsilon_ {1} epsilon_ {2}} frac {d} {ds} biggr | _ {s = 0} frac {1} { Gamma (s)} int_ {0} ^ { infty} t ^ {s-3} e ^ {itx} dt$$
I don't know how valid this is $$epsilon t = mathcal {O} (1)$$ in the big $$t$$ region, but that's all I could think of to do for now.

From there, I noticed that the integral looked a lot like the gamma function. I tried to change variables to convert it to something involving the gamma function, but the integration limits were problematic for me.

Then I tried to take the derivative inside the integral. I found it $$left ( frac {t ^ {s}} { Gamma (s)} right) & # 39; biggr | _ {s = 0} = – 1$$, resulting in the following expression:
$$frac {1} { epsilon_ {1} epsilon_ {2}} int_ {0} ^ { infty} t ^ {- 3} e ^ {itx} dt$$
It sounds relatively simple, but the evaluation of the antideriavtive gave an expression involving triginometric integrals which are divergent to zero. It makes me think that maybe one of my approximations is not valid.

I am quite certain that the resulting expression should be $$frac {1} {2 epsilon_ {1} epsilon_ {2}} x ^ {2} ( log (x) – frac {3} {2})$$, and would really appreciate a little help to prove it.

## calculation and analysis – Integral $int_ {d_1} ^ {d_2} int _ {- L / 2} ^ {L / 2} int _ {- L / 2} ^ {L / 2} frac {1} {(x ^ 2 + y ^ 2 + z ^ 2) ^ 3} dx dy dz$

I am trying to integrate the following integral in Mathematica, but it seems that it does not return a closed analytical form, nor numbers when I give values ​​for both $$d_ {1,2}$$ and $$L$$.

$$int_ {d_1} ^ {d_2} int _ {- L / 2} ^ {L / 2} int _ {- L / 2} ^ {L / 2} frac {1} {(x ^ 2 + y ^ 2 + z ^ 2) ^ 3} dx dy dz$$

Is there a tip that might be helpful in this case?
Thank you!

Better,
Denis

## calculation and analysis – How do I get this integral in the calculation

I am trying to calculate the following expression and get an analytical response but Mathematica ends up spitting out the input expression. Hoping that someone can shed some light on this.

{Integrate ((0.5 * exp ^ (- (y ^ 2 / (2 * sigy ^ 2)) – (exp ^ (i * y) – z) ^ 2 / (exp ^ (2 * i)y)(2 * sigx ^ 2))))) / (exp ^ (iy)(pi ^ 1.sigxsigy)), {y, -pi, pi})}

Attach the traditional integration expression below

## probability – Expected and integral value and sum of exchange

Consider $$E (e ^ {tX}) = int_0 ^ { infty} e ^ {tx} f (x) dx$$
f is an arbitrary probability density function

I can write $$e ^ {tx}$$ in his series. So i want to exchange the full and the sum
I thought of Lebesgue's theorem to permute the integral and the sum.
I need an integrable function which is an upper limit for the integrand. But how do you find such a function?

## calculation – Surface Integral – parametric expression

Hello. I would like to ask you for your contribution to the following problem:

Let the surface $$S$$ which surrounds the following three-dimensional domain:
$${x ^ 2 + y ^ 2≤4.0 le z le1 } text {in} Bbb R ^ 3$$
and leave $$F = (x, – y, z ^ 2-1)$$.
I would like to calculate the surface integral of $$F$$. I fully know the steps to follow, but I had trouble expressing the above domain parametrically.

## calculation and analysis – Does Mathematica return a bad value for this integral?

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## calculation – Double integral with minimum and maximum

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## real analysis – Integral of $x & # 39; mapsto e ^ {- frac {1} { alpha} | x & # 39; -x | _p}$ on a $ell_p$ -ball around $x$ in $mathbb R ^ n$
Let $$n in mathbb N ^ *$$, $$p in (1, infty)$$, $$alpha> 0$$, and $$r ge 0$$. For $$x in mathbb R ^ n$$, let $$B_ {n, p} (x; r): = {x & # 39; in mathbb R ^ n mid | x & # 39; -x | _p le r }$$ Be the $$ell_p$$-ball around $$x$$ radius $$r ge 0$$ in $$mathbb R ^ n$$.
Question. What is the value of $$u (n, p, r, alpha): = int_ {B_ {n, p} (x; r)} e ^ {- frac {1} { alpha} | x & # 39; -x | _p} dx$$ and the value (if $$p < infty$$) of $$v (n, p, r, alpha): = int_ {B_ {n, p} (x; r)} e ^ {- frac {1} { alpha} | x & # 39; -x | _p ^ p} dx$$ ?
$$int_ {0} ^ { infty} x J_0 (cx) K_v (ax) / K_v (bx) dx$$