integration – Deterministic Approximation of Products of Integrals

Suppose that I am interested in approximating the product of two integrals. For simplicity, let $P$ be a probability density function, let $f, g$ be two $P$-integrable functions, define

begin{align}
phi = int P(x) f(x) , dx \
gamma = int P(x) g(x) , dx,
end{align}

and say that I am interested in approximating the product $phi cdot gamma$.

I am interested in how a numerical analyst would approach this problem. A naive solution to this would be to do e.g. Gaussian quadrature with $P$ as the base measure, use the GQ nodes to form approximations of $phi, gamma$, and then multiply the approximations together. Can / does one ever do fancier things than this in practice?

I ask because I come from a statistics / applied probability background, where one often places a premium on having unbiased estimates of integrals. As such, taking the product of two estimates which use the same set of function evaluations will generally induce a statistical bias, and so there are simple and popular techniques which allow for this bias to be removed. I am curious about whether similar approaches are considered in numerical analysis.

To compress the question further: from the point of view of numerical analysis, are there better ways of approximating a product of integrals than taking the product of approximations to each of the integrals?

definite integrals – How to prove the converge of a sequence?

Let $t_{n} = int_{1}^{n} (ln(x)^p) ,dx$, where $p < 0$ is any real number.

How can I prove that this sequence converges?

My attempt: Clearly, $t_{n} geq 0$, so by linearly property of riemann integrals $t_{n}$ is increasing. Then, if $t_{n}$ is bounded, it is also convergent. Can you find a bound?

Why are these two integrals equal?

Why are these two equal?

$$int_{-inf}^{inf} psi (x)x frac{d}{dx}psi^{*} (x)dx = -int_{-inf}^{inf}frac{d}{dx}(psi(x)x)psi(x)^* dx $$
Found these as a part of a solution for a quantum physics problem. Tried to make them equal, by solving the first integral with partial integration. That did not work.

improper integrals – Rewrite $intlimits_{x=0}^infty{2sqrt{a -frac{b}{x}}{K_1}({2sqrt{a-frac{b}{x}}}){x^{M-1}}exp({-frac{x}{c}})dx}$ as non-elementary function?

Is it possible to rewrite this integral $I=intlimits_{x = 0}^infty {2sqrt {a – frac{b}{x}} {K_1}left( {2sqrt {a – frac{b}{x}} } right){x^{M – 1}}exp left( { – frac{x}{c}} right)dx}$ as non elementary function (For exaple $Ei(x)$, $Li(x)$) ?

${K_1}left( x right)$ is the modified Bessel function of the second kind.

$a,b,c$ are positive real number and $M$ is a positive integer

It is ok if someone can help me express the integral $I$ as an infinite series. I have also think of using the expansion
${e^x} = sumlimits_{k = 0}^infty {frac{{{x^k}}}{{k!}}}$ to deal with the exponential term but I cannot proceed.

Thank you for your enthusiasm !

calculus and analysis – All double integrals produce error

I am a beginner to the Mathematica community; however, for my Multivariable Calculus class I have made a number of double integrals that have worked successfully. Today, while working every single double integral I typed in (including retrying previously correct and functional ones) produces an error. All integrals I have tried have failed from simple to complex, but here is one example

Integrate(x^2*y+3*x*y^2, {x, -Sqrt(8), Sqrt(8)}, {y, 0, (8-x^2)})
produces error
“Invalid integration variable or limit(s) in {x^3, 0, 1}”

Could someone please help me with this issue?

co.combinatorics – Saddle point approximation for multiple contour integrals

General Question: Is there a reference where the saddle point approximation is applied to multiple contour integrals?

In particular, say we have the integral
$$ I_N = frac{1}{(2pi i)^N} oint left(prod_{ell=1}^N frac{dq_{ell}}{q_{ell}^2}right) left(sum_{j=1}^{N} q_{j}right)^{N},$$
where we are applying $N$ contour integrations in sequence on curves that circle the origin in the complex plane. Using the multinomial theorem one can show that $I_N = N!$.

Using elementary symmetric polynomials, one can also show that $I_N$ can be written as

begin{equation}
I_N = frac{1}{(2pi i)^{2N}} oint left(prod_{ell=1}^Nfrac{dz_{ell}}{z_{ell}^2} right) left(prod_{m=1}^Nfrac{dq_{m}}{q_{m}^2} right) prod_{i, j=1}^NBig(1 + z_i q_jBig),
label{eq:INcont}
end{equation}

where we are now applying $2N$ contour integrations.

Specific Question: Is it possible to apply a multi-dimensional saddle point approximation to either form of $I_N$ to derive some form of Stirling’s approximation (i.e., $N!simeq sqrt{2pi N} (N/e)^N$)?

(Reposted from stackexchange due to no answers.)

Convergence of definite integrals given unknowns.

Given the following equation,
$$
int_{1}^{2} frac{1}{x(ln x)^{a}},dx
$$

I am supposed to find the condition on the constant $a$ such that the above integral is convergent.

Substituting $u = ln(x)$, the above equation equates to

$$
int_{0}^{ln 2} frac{1}{u^{a}}du
$$

Which simplifies to

$$
frac{(ln 2)^{1-a}}{1-a}
$$

However, I am unsure of what values the above equation converges to.

How to simplify Elliptic integrals

Context

I would like to show that these 2 functions are identical

f1(a_) = (2/(Pi)) Integrate(Cos(t)/(1 - 2 a Cos(t) +
          a^2)^(3/2), {t, 0, (Pi)}, 
    Assumptions -> {0 < a < 1}) // FullSimplify

enter image description here

corresponding to this integral

$$ frac{2}{pi } int_0^{pi } frac{cos (theta )}{left(1-2 a cos (theta )+a^2right)^{3/2}} , dtheta $$

and

f2(a_) = (4/((Pi) a (1 - a^2)^2))((1 + a^2) *
           EllipticE(a^2)-(1 - a^2) EllipticK(a^2))

enter image description here

This does not seem to work:

f2(a)/f1(a)  // FullSimplify(#, Assumptions -> {0 < a < 1}) &

Any suggestions?

Issues with convergence of an iteration of numerical integrals computed with NIntegrate

this is my first question on Stack-Exchange. I am not an expert in numerical integration and I am having issues of convergence with some numerical integration. I am trying to solve numerically a nonlinear differential equation of which I know an exact solution. I want to compare my numerical results with the exact one (to check the correctness of the code). Unfortunately, after 2 iterations the numerical result starts departing from the exact behavior.

This is the differential equation:

y”(t)=(y(t)^2-1)y(t)

which I wrote in an integral form (I need to solve it in the integral form for other reasons) in terms of the Green function

G(t-x)=(1/2) (t-x) sign(t-x).

An analytic solution is given by

y(x)=Tanh(x/sqrt{2})

The Mathematica code I am using is the following:

    t1 = -10;
    t2 = 10;
    n = 100;
    m = 5;
    t = Range(t1, t2, (t2 - t1)/(n));
    
    Do(y(x_, 0) = Tanh(x/Sqrt(2));
    list(j) = {t, NIntegrate( (1/2)(t-x) Sign(t-x)(y(x,j-1)^2 - 1) y(x, j-1), {x, -10, 10}, 
    AccuracyGoal -> 13, WorkingPrecision -> 13)} // Transpose;
    sol(j) = Interpolation(list(j), Method -> "Spline");
    y(x_, j) = sol(j)(x)
    , {j, m})

Basically, I am trivially starting from the exact solution for j=0, and check the convergence at higher order in the iteration (I am doing this to verify whether the code is correct). But from j=3 the numerical solution starts departing from the exact one; so I would like to find a way to correct my code and make the iteration convergent for any higher j, as it should be.

I hope I have clearly stated my problem. Many thanks in advance for your help.

cv.complex variables – Real integrals with complex analysis

I don’t have a clear formal viewpoint on this problem.
Resolving the Euler-Lagrange equations for the string with a point mass perturbation:
$$ frac{partial^2 phi }{partial x^2} = delta (x-a)$$
I encountered the following integral:
$$ I = int_{ – infty}^{+ infty} dk f(x,k) = int_{ – infty}^{+ infty} dk frac{e^{ik(x-a)}}{k^2} $$
Now, during the lessons we computed it as:
$$ I = lim_{ epsilon longrightarrow 0}int_{ gamma_+ cup gamma_-} dk frac{e^{ik(x-a)}}{(k – i epsilon)^2} $$
with $gamma_+$ being the semicircle in the upper half plane, counterclockwise, and $gamma_-$ being the semicircle in the lower half plane, clockwise. We found that
$$ I = 2 pi (x-a) theta(x-a) $$
My first question is:
If we compute it with choosing another pole shift such as $(k+i epsilon)$, we have:
$$int_{ gamma_-} dk frac{e^{ik(x-a)}}{(k-iepsilon)^2} = – 2 pi (x-a) theta(a-x) $$

In general every shift can give a different result.
Is it due to the fact that
$$ lim_{epsilon longrightarrow 0 } int neq int lim_{epsilon longrightarrow 0 }$$
and I can’t commute the operations (because for instance the integrand $f(x,k,epsilon)$ isn’t dominated by a function $g(x,k)$ )???
Moreover, how can I show I can’t commute the two? And in which case I can commute?
My second question is:
Is there another method that uniquely provide an answer to the integral $I$ above?