Absolute convergence of an complex integral

I would like to show that the integral $$int_{B-iinfty}^{B+iinfty}left|at^2+frac{M}{4a}right|^{-(1+s)}|e^{2pi nt}|dt$$ converges for $Re(s)>-frac{1}{2}$,B>0, $ain mathbb{Z}, n,Min mathbb{N}$ and satisfies an uniform estimate. Unfortunately, I do not know how to start, because I did not find anything about the convergence of a complex integral. Furthermore, I am not sure what is meant by uniform estimate meaning from what is the estimate independent. Note that I do not need the evaluation of the integral. I am thankful for every help and hint.

Line Integral Motivation

Is there a case to be made that the topic of line integrals should only involve vector fields?

My colleagues and our textbook take the position that line integrals should only be taught from a vector field perspective (specifically for calculating “work”). [In fact, our textbook defines a line integral as $int _C mathbf{F} cdot dmathbf{r}$, where $mathbf{F}$ is a vector field and $C$ is some parametrized curve.]

I think it makes more pedagogical sense to introduce line integrals as a way to generalize what students should have just done in their integral calculus class: integrate a function along some 1-dimensional direction. Now, that “direction” can be a path in 2-space or 3-space, so we can see an area as the result of the line integral. Then, after introducing vector fields, we can consider other, meaningful things to integrate along a path, such as a dot product of the field with the path.

My motivation here is that I would like new calculus topics to be easily connected to old topics, if possible. If we jump right into calculating work without any tie back to the “calculate the area” problem students are used to, I fear they may come away thinking that line integrals are just these weird things with their own rules.

In short: Is there a prevailing setting for introducing line integrals? If so, has there been a movement to minimize the teaching of line integrals over scalar fields, focusing primarily on work calculations?

Evaluate over a two-dimensional domain, the integral of (hypergeometric-based) f(d,k), the result for f(d,0)…

I view this as both a mathematics and a Mathematica question–so apologies if it is thought I should have sent it alternatively to the mathematics stack exchange.

I want to perform the two-dimensional integration (or, possibly, reduce to a one-dimensional integration)

Integrate[Y^(-1 + d)
   Hypergeometric2F1Regularized[d/2, -k, (2 + d)/
   2, ((Y^2 - e^2 Subscript[r, 14]^2) (-1 + e^2 Subscript[r, 14]^2))/(
   e^2 (Y^2 - Subscript[r, 14]^2) (-1 + Subscript[r, 14]^2))] (1/(
   e Subscript[r, 14]))^(
  1 + d) (1 + Y^2 (1 - 1/Subscript[r, 14]^2) - Subscript[r, 14]^2)^
  k (1 - e^2 Subscript[r, 14]^2)^(
  d/2) (-Y^2 + e^2 Subscript[r, 14]^2)^(d/2), {Subscript[r, 14], 0, 
  1}, {Y, e Subscript[r, 14]^2, e Subscript[r, 14]}, 
 Assumptions -> d >= 1 && k >= 0 && 0 < e <= 1]

So, $d$ and $k$ are parameters, and $Y$ and $r_{14}$ the variables of integration, with $e$ being a free variable.

In TeX, the integrand is the product of
begin{equation}
Y^{d-1} left(frac{1}{r_{14} epsilon }right){}^{d+1} left(1-r_{14}^2 epsilon
^2right){}^{d/2} left(left(1-frac{1}{r_{14}^2}right) Y^2-r_{14}^2+1right){}^k
left(r_{14}^2 epsilon ^2-Y^2right){}^{d/2}
end{equation}
and
begin{equation}
, _2tilde{F}_1left(frac{d}{2},-k;frac{d+2}{2};frac{left(r_{14}^2 epsilon
^2-1right) left(Y^2-r_{14}^2 epsilon ^2right)}{left(r_{14}^2-1right) epsilon ^2
left(Y^2-r_{14}^2right)}right) .
end{equation}
The two-dimensional domain of integration is
begin{equation}
r_{14} in [0,1], hspace{.25in} Y in [varepsilon r_{14}, varepsilon^2 r_{14}] .
end{equation}

For $k=0$, the integral evaluates (as can be confirmed by setting $d$ to
a positive integer--even integers evaluate more readily) to

1/4 e^(-1 + d)
  Gamma[d/2] Gamma[
  d] HypergeometricPFQRegularized[{-(d/2), d/2, d}, {1 + d/2, 
   1 + (3 d)/2}, e^2]

That is,
begin{equation}
frac{1}{4} epsilon ^{d-1} Gamma left(frac{d}{2}right) Gamma (d) ,
_3tilde{F}_2left(-frac{d}{2},frac{d}{2},d;frac{d}{2}+1,frac{3 d}{2}+1;epsilon
^2right).
end{equation}

I've been trying all possible integration-by-parts combinations with no success to this point in time.

For even, nonnegative $d$ and nonnegative $k$, the integrand evaluates to a polynomial in $e$.
For odd $d$, logs and polylogs appear.

The question stated here pertains to the issue discussed in sec. IX.B of my posting,
https://arxiv.org/abs/1803.10680, "Qubit-qudit separability/PPT-probability investigations, including Lovas-Andai formula advancements",
of finding an “extended Lovas-Andai master formula”, denoted there by $ tilde{chi}_{d,k}(varepsilon)$.

Trouble solving the integral with mathematica

I have the following function:

(1/((x - y)^2 + (z - [Zeta])^2))(c[Alpha] (x - y)^2 + 
   cz (z - [Zeta])^2) (((x - 
      y)^2 (z^2 (x^2 [ScriptA][ScriptA] + 
         x [ScriptA][ScriptB] + [ScriptA][ScriptC]) + 
      x^2 [ScriptM] - y^2 [ScriptM] + x [ScriptN] - y [ScriptN] + 
      z (x^2 [ScriptS] + 
         x [ScriptT] + [ScriptU]) - (y^2 [ScriptS] + 
         y [ScriptT] + [ScriptU]) [Zeta] - (y^2 [ScriptA]
[ScriptA] + 
         y [ScriptA][ScriptB] + [ScriptA][ScriptC]) [Zeta]^2))/
   Sqrt[(x - y)^2 + (z - [Zeta])^2] + ((x - 
      y) (z - [Zeta]) (z^2 (x^2 [ScriptA][ScriptD] + 
         x [ScriptA][ScriptE] + [ScriptA][ScriptF]) + 
      x^2 [ScriptP] - y^2 [ScriptP] + x [ScriptQ] - y [ScriptQ] + 
      z (x^2 [ScriptV] + 
         x [ScriptW] + [ScriptY]) - (y^2 [ScriptV] + 
         y [ScriptW] + [ScriptY]) [Zeta] - (y^2 [ScriptA]
[ScriptD] + 
         y [ScriptA][ScriptE] + [ScriptA][ScriptF]) [Zeta]^2))/
   Sqrt[(x - y)^2 + (z - [Zeta])^2] + (1/
   Sqrt[(x - y)^2 + (z - [Zeta])^2])(x - 
      y) ((x - 
         y) (z^2 (x^2 [ScriptA][ScriptA] + 
            x [ScriptA][ScriptB] + [ScriptA][ScriptC]) + 
         x^2 [ScriptM] - y^2 [ScriptM] + x [ScriptN] - 
         y [ScriptN] + 
          z (x^2 [ScriptS] + 
            x [ScriptT] + [ScriptU]) - (y^2 [ScriptS] + 
            y [ScriptT] + [ScriptU]) [Zeta] - (y^2 [ScriptA]
 [ScriptA] + 
            y [ScriptA][ScriptB] + [ScriptA][ScriptC]) [Zeta]^2) 
 + (z - [Zeta]) (z^2 (x^2 [ScriptA][ScriptD] + 
            x [ScriptA][ScriptE] + [ScriptA][ScriptF]) + 
         x^2 [ScriptP] - y^2 [ScriptP] + x [ScriptQ] - 
         y [ScriptQ] + 
         z (x^2 [ScriptV] + 
            x [ScriptW] + [ScriptY]) - (y^2 [ScriptV] + 
            y [ScriptW] + [ScriptY]) [Zeta] - (y^2 [ScriptA]
[ScriptD] + 
            y [ScriptA][ScriptE] + [ScriptA][ScriptF]) 
[Zeta]^2)))

Where the four variables are ${x,y,z,zeta}$; ${c_{alpha},c_z} $ are two constants; ${m,n,o,p,q,…,aa,ab,ac…}$ are 18 constant parameters. I have to integrate this quantity and decided to it in principal values:

   E[1]=Integrate[
 Integrate[[ScriptCapitalE] UnitBox[(y - L/2)/
    L] UnitBox[([Zeta] - h/2)/h], y], [Zeta]]

and then

P=([DoubleStruckCapitalE][1] /. {[Zeta] -> z - [CurlyEpsilon], 
    y -> x - [CurlyEpsilon]}) - ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z - [Delta]z, 
    y -> x - [CurlyEpsilon]}) - ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z - [CurlyEpsilon], 
    y -> x - [Delta][Alpha]}) + ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z - [Delta]z, 
    y -> x - [Delta][Alpha]}) + ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z + [Delta]z, 
    y -> x - [CurlyEpsilon]}) - ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z + [CurlyEpsilon], 
    y -> x - [CurlyEpsilon]}) - ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z + [Delta]z, 
    y -> x - [Delta][Alpha]}) + ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z + [CurlyEpsilon], 
    y -> x - [Delta][Alpha]}) + ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z + [Delta]z, 
    y -> x + [Delta][Alpha]}) - ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z + [CurlyEpsilon], 
    y -> x + [Delta][Alpha]}) - ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z + [Delta]z, 
    y -> x + [CurlyEpsilon]}) + ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z + [CurlyEpsilon], 
    y -> x + [CurlyEpsilon]}) + ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z - [CurlyEpsilon], 
    y -> x + [Delta][Alpha]}) - ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z - [Delta]z, 
    y -> x + [Delta][Alpha]}) - ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z - [CurlyEpsilon], 
    y -> x + [CurlyEpsilon]}) + ([DoubleStruckCapitalE][
    1] /. {[Zeta] -> z - [Delta]z, y -> x + [CurlyEpsilon]})

I then have to integrate all of this in ${x,y}$:

 Integrate [Integrate[P, {z, 0, h}], {x, 0, L}]

From all of this, I get a function of the unknown coefficients as a result. This function has Complex numbers. Something must be wrong in my procedure since we know how Riemman integration of a real function leads to a real result

Integral of pdf and cdf normal standard distribution

$ int_{-infty}^{infty}N(a+bz_m)n(z_m)dz_m=Nleft(frac{a}{sqrt{1+b^2}}right) $

I have a problem with that theorem. I’ve tried to proof it by calculate the pdf of $ n(a+bz_m) $ first and combine it wit the pdf of $ n(z_m) $ but the result is complicated and doesn’t yield $ nleft(frac{a}{sqrt{1+b^2}}right) $ . I really need your help. Thanks.

Inner Product, Definite Integral

Does the map $$ $=$ $int _0^1:left(left(fleft(xright)-frac{d}{dx}fleft(xright)right)left(gleft(xright)-frac{d}{dx}gleft(xright)right)right)dx$ define an inner product on the set of all polynomial functions of order less than or equal to $n$?

Although I felt that this map was symmetric and bilinear by the properties of definite integrals, I did not think it was positive definite. This was because, say, $$ would become $int _0^1:left(left(fleft(xright)-frac{d}{dx}fleft(xright)right)^2right)dx$, which has value greater than zero for values other than the zero function. For example, if $f(x)$ was $2x$, then $f'(x)$ would be $2$ and the value of the above definite integral would be $4/3$. This is why I don’t think this space defines an inner product.

So what I am worried about is that I am misunderstanding how the concept of zero vectors with regards to inner products work as this seems to obvious.

If anyone can verify what I have done or tell me how I went wrong, I would greatly appreciate it!

Integral Inequality with L-2 Norm

On page 135 of The Mathematical Theory of Finite Element Methods (Brenner and Scott), I encountered the following inequality:

$left | int_{Gamma} overline{v} – v , ds right | leq |Gamma |^{1/2} || overline{v} – v ||_{L^2(partial Gamma)},$.

The mean $overline{v}$ is defined as

$overline{v} = frac{1}{text{meas}(Omega)} int_{Omega} v(x) , dx ,$

and $Gamma = partial Omega$.

The inequality is part of a proof that the bilinear form for the Poisson equation with Dirichlet boundary conditions is coercive.

My question is: could I replace the $overline{v}-v$ on both sides of the inequality with some (more general) function? If so, what would the restrictions on the function be? I’m not sure where the inequality comes from.

Generalize Jensen’s Integral Inequality to the product of two functions

Let $E$ be a measurable set with $m(E)>0$.

Let $f$, $gamma$ be two measurable, real-valued function which are finite a.e. on $E$ with $f, gamma$ and $fcdot gamma$ all integrable.

Assume $gammageq 0$ and $int_{E}gamma>0$. Then if $phi$ is a convex function on an open interval $(a,b)$ containing the range of $f$, then we have

$phiBig(dfrac{int_{E}fcdotgamma}{int_{E}gamma}Big)leqBig(dfrac{int_{E}phi(f)cdotgamma}{int_{E}gamma}Big)$

I am trying to modify the proof of the simpler version (the one with only $f$) of Jensen’s inequality to this more general version, but I don’t really know how to deal with the new function $gamma$.

Any hints or detailed explanations are really appreciated!!

Integral converges only if Mathematica kernel has just started?

After just starting Mathematica (or quitting the kernel) if I run:

ClearAll[r, m, e]
e = Integrate[(Sinc[p r] - 1) p^2/(p^2 + m^2), {p, 0, Infinity}];
Assuming[ r > 0 && m > 0, Simplify[e]]

I get:

1/2 [Pi] (m + E^(-m r)/r)

but it seems that if I happen to evaluate this cell again, I get a non-convergent integral error:

error output

I am using Mathematica 11.2.0 for mac. I don’t see anything stateful in this small calculation that would make the integral converge one time, but not the other — any idea what’s going on?