## analytical number theory – Contour integration involving the Zeta function

I am trying to calculate the integral of the contour
$$frac {1} {2 pi i} int_ {c – i infty} ^ {c + i infty} zeta ^ 2 ( omega) frac {8 ^ omega} { omega} omega$$
or $$c> 1$$, $$zeta (s)$$ is Riemann's zeta function.

Use Perron's formula and define $$D (x) = sum_ {k leq x} sigma_0 (n)$$, or $$sigma_0$$ is the usual function of counting the divisors, we can show that
$$D (x) = frac {1} {2 pi i} int_ {c – i infty} ^ {c + i infty} zeta ^ 2 ( omega) frac {x ^ omega } { omega} d omega.$$
So for this purpose, we can just calculate $$D (8)$$ and call it a day. However, for my own needs, I want to redefine $$D (x)$$ by the full above instead. Therefore, why I state the problem for a specific case $$x = 8$$, for example.

Considering a modified Bromwich contour which avoids branch cutting and $$z = 0$$, let's call it $$mathcal {B}$$, we can apply the Cauchy residue theorem:
$$oint _ { mathcal {B}} zeta ^ 2 ( omega) frac {8 ^ omega} { omega} d omega = 2 pi i operatorname * {Res} ( zeta ^ 2 ( omega) frac {8 ^ { omega}} { omega}; 1) = 8 (-1 + 2 gamma + ln 8)$$
or $$gamma$$ is the Euler-Mascheroni constant. I got this by extending $$zeta ^ 2 ( omega) frac {8 ^ omega} { omega}$$ in his Laurent series. To obtain the desired integral, it would then be necessary to subtract from this value the parts of the contour which are not the vertical line $$c – iR$$ at $$c + iR$$, subtract them from the residual value obtained, then take the limit as $$R to infty$$ and $$r to 0$$ or $$C_r$$ is the circle of radius $$r$$ where the $$mathcal {B}$$ dodges the origin.

Feel free to modify this outline in any shape or form, or consider a positive integer value different from $$x$$.

## calculation and analysis – Numerical solution of triple integration of a region with variable bounds?

I am asked to calculate the mass of the region bounded by the plane `(2x + 3y - z = 2)` and below by the triangle on the xy plane with vertices `(0,2),(1,0),(4,0)`. The density is proportional to the distance from the plane to the xy plane, so the function is `d(x,y,z) = kz`.

My limits are:

`0 ≤ z ≤ (2x + 3y - 2)`

`(1 - 0.5y) ≤ x ≤ (4 - 2y)`

and `0 ≤ y ≤ 2`.

My code is:

``````Integrate(d(x,y,z),{z,0,2x+3y-2},{x,1-0.5y,4-2y},{y,0,2})
``````

but Mathematica does not return a number (if I'm not mistaken, it should be 19k).

Instead, I get: `k(3-1.5y)(-2+2x+3y)^2`.

However if I integrate the function step by step:

``````Integrate(Integrate(Integrate(d(x,y,z),{z,0,2x+3y-2}),{x,1-0.5y,4-2y}),{y,0,2}),
``````

I get a numeric value.

Is there some syntax that Mathematica requires that I am not aware of?

## calculation and analysis – How to manage the units during integration?

I have defined some functions, which are to use units. Now another function is used to integrate with them, with the integration variable also quantized and the input of said function. I better show code:

It is not pretty, but I feel like nothing is pretty in Mathematica. I am not very proficient in the Wolfram language, and I don't have a lot of experience using symbolic expressions.

Back to the code: it will work well, but when I plug in numbers or variables, it does not work:

So, as you can see, something is wrong. I did several hours of research yesterday and I just can't seem to get it to work. I want to plot this and do some calculations, so nothing special, but for it to work, it has to accept inputs like `1 s` and `t s`.

Bonus: I cannot understand my delayed duties. As you can see, h (x, s, t) evaluates well to an easy piecewise function. So why does it take ages and millennia to trace this? Do I have to `h(x_, s_, t_) = ...` instead of `:=`? It didn't seem to do anything for me. Also, this assumes that I haven't used units, for it to work.

The code in the text:

``````Clear("Global`*")
g = UnitConvert(
Entity("Planet", "Earth")(EntityProperty("Planet", "Gravity")));
TWR(t : Quantity(_, unit_)?(CompatibleUnitQ(#, "Seconds") &)) :=
FullSimplify(
Piecewise({{0.65 Quantity(1, (1/("Seconds")))*t,
0 Quantity(1, "Seconds") < t <
2.05 Quantity(1, "Seconds")}, {1.3325,
t >= 2.05 Quantity(1, "Seconds")}}), t (Element) Reals)
a(t : Quantity(_, unit_)?(CompatibleUnitQ(#, "Seconds") &)) :=
FullSimplify((TWR(t) - 1) g, t (Element) Reals)
v(t : Quantity(_, unit_)?(CompatibleUnitQ(#, "Seconds") &)) :=
FullSimplify(
Integrate(
a(Quantity(T, "Seconds")), {Quantity(T, "Seconds"),
Quantity(0, "Seconds"), t}, Assumptions -> t (Element) Reals),
t (Element) Reals)
h(x_, s_, t : Quantity(_, unit_)?(CompatibleUnitQ(#, "Seconds") &)) :=
FullSimplify(
x Quantity(1, "Meters") +
s Quantity(1, (("Meters")/("Seconds")))*t +
Integrate(
v(Quantity(c, "Seconds")), {Quantity(c, "Seconds"),
Quantity(0, "Seconds"), t},
Assumptions -> t (Element) Reals && c (Element) Reals),
t (Element) Reals)
``````

## integration – Moment of inertia using center of mass as origin

I am trying to determine the inertia tensor of a rectangular block. I will later implement this inertia tensor in the dynamic equation of a satellite body frame, which means that the origin must be on the center of mass of the block. I looked at examples of rectangular blocks and the only ones I could find were the origin of the coordinate frame on one edge of the block, and I couldn't find how to translate them into a coordinate system based on the center of gravity.

As the body is symmetrical, the non-diagonal elements of the inertia tensor would obviously be 0. That said, I have calculated the diagonal elements having the frame at the edge of the block using:

$$I_ {xx} = frac {m} {3} (y ^ 2 + z ^ 2); I_ {yy} = frac {m} {3} (x ^ 2 + z ^ 2); I_ {zz} = frac {m} {3} (x ^ 2 + y ^ 2)$$

which comes from the integral:

$$I = int_m (y ^ 2 + z ^ 2) dm$$

So how can I calculate these moments of inertia using the original center of mass? Is it as simple as halving the distances? Or is there more?

## 2013 – Integration of Azure (or IRM) SP2013 information protection

I am looking for a way to tag documents in SharePoint 2013 with Microsoft AIP.
Looking online, I can't find anything related to this specific use case:

Open the document form page in SharePoint (not the document itself) and select a field value that will label the document.

OR

Tag a document using web services.

From my research, it doesn't seem possible, but it doesn't hurt to ask.

## Web application – Integration procedure or let them explore?

Work on improving the application integration process. It's a great app with multiple modules (think Atlassian), and I can't decide whether or not start users on the main dashboard to let them choose a module in the left menu, or to have them choose before entering the platform. Thoughts? Does data support one way or another?

## Integration ux – How to collaborate and define a new style guide for an existing business design?

As a naïve designer, I have a hard time defining a style guide. There is a style guide created with Adobe XD. As a team effort, we would like to improve and manage the style guide. A reasonable and productive solution is to use Frontify. Unfortunately, the company is running out of budget and we have to look for an alternative that can provide the same functionality as Frontify.

Anyone facing the same problem?

## css – About the responsiveness of posts integration on Instagram

I have inserted integrations on my personal page. I want the page to be responsive. Once I've shrunk the screen, there comes a point when Instagram windows stop shrinking, until they touch the edge of the screen, and then they shrink again, but this time on the edge of the screen, it looks ugly.

How, by modifying the code, can I make these windows stay responsive as they were originally and always shrink in proportion to the size of the screen?

I program in HTML5, with CSS3.

## saml – SimpleSAML SSO PHP integration

In my business, I have an existing website built with Laravel, and they want me to use SimpleSaml for SSO connections with an existing IDP. I am very new to PHP because I am a .NET developer and I would like to know how to do it (first steps for a beginner with PHP).

I only received metadata. I managed to make the website work locally using `php artisan serve`

``````\$metadata('https://sit-sso-nccd.esa.edu.au/simplesaml/saml2/idp/metadata.php') = array (
'SingleSignOnService' =>
array (
0 =>
array (
'Binding' => 'urn:oasis:names:tc:SAML:2.0:bindings:HTTP-Redirect',
'Location' => 'https://sit-sso-nccd.esa.edu.au/simplesaml/saml2/idp/SSOService.php',
),
),
'SingleLogoutService' =>
array (
0 =>
array (
'Binding' => 'urn:oasis:names:tc:SAML:2.0:bindings:HTTP-Redirect',
'Location' => 'https://sit-sso-nccd.esa.edu.au/simplesaml/saml2/idp/SingleLogoutService.php',
),
),
'NameIDFormat' => 'urn:oasis:names:tc:SAML:2.0:nameid-format:transient',
'contacts' =>
array (
0 =>
array (
'contactType' => 'technical',
),
),
);
``````

``````\$metadata('moodle') = array (
'SingleLogoutService' =>
array (
0 =>
array (
'Binding' => 'urn:oasis:names:tc:SAML:2.0:bindings:HTTP-Redirect',
'Location' => 'https://sit-plm-nccd.esa.edu.au/simplesaml/module.php/saml/sp/saml2-logout.php/moodle-sp',
),
1 =>
array (
'Binding' => 'urn:oasis:names:tc:SAML:2.0:bindings:SOAP',
'Location' => 'https://sit-plm-nccd.esa.edu.au/simplesaml/module.php/saml/sp/saml2-logout.php/moodle-sp',
),
),
'AssertionConsumerService' =>
array (
0 =>
array (
'index' => 0,
'Binding' => 'urn:oasis:names:tc:SAML:2.0:bindings:HTTP-POST',
'Location' => 'https://sit-plm-nccd.esa.edu.au/simplesaml/module.php/saml/sp/saml2-acs.php/moodle-sp',
),
1 =>
array (
'index' => 1,
'Binding' => 'urn:oasis:names:tc:SAML:1.0:profiles:browser-post',
'Location' => 'https://sit-plm-nccd.esa.edu.au/simplesaml/module.php/saml/sp/saml1-acs.php/moodle-sp',
),
2 =>
array (
'index' => 2,
'Binding' => 'urn:oasis:names:tc:SAML:2.0:bindings:HTTP-Artifact',
'Location' => 'https://sit-plm-nccd.esa.edu.au/simplesaml/module.php/saml/sp/saml2-acs.php/moodle-sp',
),
3 =>
array (
'index' => 3,
'Binding' => 'urn:oasis:names:tc:SAML:1.0:profiles:artifact-01',
'Location' => 'https://sit-plm-nccd.esa.edu.au/simplesaml/module.php/saml/sp/saml1-acs.php/moodle-sp/artifact',
),
),
'contacts' =>
array (
0 =>
array (
'contactType' => 'technical',
),
),
);
``````

## computation and analysis – The order of integration in indefinite integral subjects?

``````Clear("Global`*")

f = (-(x1 - x11)^4 - 6 (x1 - x11)^2 (x2 - x22)^2 +
3 (x2 - x22)^4)/(4 Pi ((x1 - x11)^2 + (x2 - x22)^2)^3);
``````

An indefinite integral, that is to say, `antiderivative` is not unique

``````ad1 = FullSimplify@Integrate(f, x1, x2, x11, x22)

(* (1/(16 (Pi)))(3 (x1 - x11)^2 -
8 (x1 - x11) (x2 - x22) ArcTan((x1 - x11)/(
x2 - x22)) + (-3 (x1 - x11)^2 + (x2 - x22)^2) Log((x1 - x11)^2 + (x2 -
x22)^2)) *)
``````

Check that `ad1` East a valid anterivative of `f`

``````f == D(ad1, x1, x2, x11, x22) // Simplify

(* True *)

ad2 = FullSimplify@Integrate(f, x22, x11, x2, x1)

(* (1/(16 (Pi)))(-(x2 - x22)^2 +
8 (x1 - x11) (x2 - x22) ArcTan((x2 - x22)/(
x1 - x11)) + (-3 (x1 - x11)^2 + (x2 - x22)^2) Log((x1 - x11)^2 + (x2 -
x22)^2)) *)
``````

Check that `ad2` East a valid anterivative of `f`

``````f == D(ad2, x22, x11, x2, x1) // Simplify

(* True *)
``````