Calculate $ int _ { Gamma} omega $ when $ omega = z (z-y) dx + xzdy-xydz $, $ Gamma = Gamma_1 cup Gamma_2 cup Gamma_3 $

Calculate $ int _ { Gamma} omega $ when $ omega = z (z-y) dx + xzdy-xydz $

$ Gamma = Gamma_1 cup Gamma_2 cup Gamma_3 $

$$ x ^ 2 + y ^ 2 = (z-1) ^ 2 $$

$ x geq0, y geq0, z geq0 $

$ Gamma_ {1,2,3} $ are intersections with planes $ x = $ 0, $ y = 0 $, $ z = $ 0

How to find the right setting?

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Evaluation: 5

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$ alpha + beta = sup_ {v < gamma} ( alpha + beta_ {vb}) $?

On page 124 of Introduction to Set Theory, Jech states that the ordinal functions below are continuous in the second variable: If $ gamma $ is a limit ordinal and $ beta = sup_ {v < gamma} beta_ {vb} $ then
$$ alpha + beta = sup_ {v < gamma} ( alpha + beta_ {vb}) $$
$$ alpha. beta = sup_ {v < gamma} ( alpha. beta_ {vb}) $$
$$ alpha ^ beta = sup_ {v < gamma} ( alpha ^ { beta_ {vb}} $$
Then, we say that the above results are tracked directly from the definitions!
Someone has answered something that asks the same question but the above problem, taken directly from Jech's book, is formulated slightly differently by considering the indexes and this has caused confusion in my mind since I am new to the subject. I would be grateful if you could clarify the exact arguments above.

complex analysis – Calculate $ int_ {0} ^ {+ infty} frac { sin x} { sqrt {x}} dx $ using the Gamma function

I want to calculate
$$ int_ {0} ^ {+ infty} enac { sin x} { sqrt {x}} dx $$
using the gamma function.

I know that by variable change, $ y = sqrt {x} $, we obtain
$$ int_ {0} ^ {+ infty} frac { sin x} { sqrt {x}} dx = 2 int_ {0} ^ {+ infty} sin y ^ 2 dy = frac { sqrt {2 pi}} {2} $$
by the Fresnel Integral.

I try it by considering this:
$$ int_ {0} ^ {+ infty} x ^ {- frac {1} {2}} e ^ {ix} dx $$
It converges for both real and imaginary parts using the Dirichlet test, and $ 0 It's not a problem here. Let the square root take the main branch where $ sqrt {1} = $ 1. Let $ y = -ix $then
$$ int_ {0} ^ {+ infty} x ^ {- frac {1} {2}} e ^ {ix} dx = sqrt {i} int_ {0} ^ {- i infty } y ^ {- frac {1} {2}} e ^ {- y} dy = ( frac { sqrt {2}} {2} + frac { sqrt {2}} {2} i ) Gamma ( frac {1} {2}) = frac { sqrt {2}} {2} + frac { sqrt {2 pi}} {2} i $$
And that coincides with the final answer!

My problem is, suppose $ L $ is a radius from $ 0 and has an angle $ phi $ with the $ x $-axis, and leave $ phi in (0.2 pi) $.
I want to argue that (maybe that's incorrect though)
$$ Gamma (z) = int_ {L} t ^ {z-1} e ^ {- t} dt $$

I know first that it converges when $ Re (z)> 0 $. Choose an outline as the area, leave $ L_1 = {z = x + iy: y = 0, r <x <R } $, $ L_2 = {z = Re ^ {i theta}: 0 < theta < phi $, $ L_3 = {z = xe ^ {i phi}: r <x <R } $ and $ L_4 = {z = re ^ {i theta}: 0 < theta < phi $, or $ r <R $ and the outline is counterclockwise. According to Cauchy's theorem, the contour integral must be $ 0.
Easy to see that $ z = x + iy $)
$$
lim_ {r to0 +, R to + infty} int_ {L_1} t ^ {z-1} e ^ {- t} dt = Gamma (z)
$$

$$
| int_ {L_4} t ^ {z-1} e ^ {- t} dt | = | int _ { phi} ^ {0} e ^ {- re} (re ^ {i theta}) ^ {z-1} ire ^ {i theta} d theta | leq int_ {0} ^ { phi} e ^ {- r cos theta} | r ^ {z} e ^ {i theta (z-1)} | d theta = int_ {0} ^ { phi} e ^ {- r cos theta} r ^ {x} e ^ {- theta y} d theta to 0, r to 0 +
$$

But when we consider $ L_2 $:
$$
| int_ {L_2} t ^ {z-1} e ^ {- t} dt | leq int_ {0} ^ { phi} e ^ {- R cos theta} R ^ {x} e ^ {- theta y} d theta
$$

when for example, $ frac {3 pi} {2}> phi> frac { pi} {2} $, we have $ cos theta <0 $ and I failed to prove that the integral above goes to zero when $ R to + infty $.

Is my use of the Gamma function to calculate the original integral a coincidence to get the correct result or is there a way to prove my argument?

Thank you so much!

sequences and series – prove $ sum_ {k = 2} ^ { infty} frac { log k} {k ^ 2} = frac { pi ^ 2} {6} big (12 log A- gamma- log2 pi) $

I just found this bad boy on Wikipedia and I was instantly in love
$$ sum_ {k geq2} frac { log k} {k ^ 2} = frac { pi ^ 2} {6} big (12 log A- gamma- log2 pi) $ $
Right here $ A $ refers to the Glaisher-Kinkelin constant.

Here is what I have tried until now.

To define
$$ S = sum_ {k geq2} frac { log k} {k ^ 2} $$
Recall $$ zeta (s) = sum_ {k geq1} frac1 {k ^ s} $$
taking $ d / ds $ on both sides,
$$ zeta (s) = – sum_ {k geq1} frac { log k} {k ^ s} $$
$$ S = – zeta (2) $$
But I am not satisfied! How do I proceed? Thank you.

Analytical Geometry – The value of $ alpha + beta + gamma $ is ______.

A variable line $ ax + by + c = 0 $ where a, b, c are in AP, is normal to a circle $ (x- alpha) ^ 2 + (y- beta) ^ 2 = gamma $which is orthogonal to the circle $ x ^ 2 + y ^ 2-4x-4y-1 = 0 $. The value of $ alpha + beta + gamma $ is _______.

Attempt

The normal goes through the center of the circle so_$ a alpha + b beta + c = 0 $.

The condition of AP also led to: $ 2a alpha + a beta + c beta + 2c = 0 $

The condition of the orhtogonal cicles gave the equation:
$ 15 { alpha} ^ 2 + 15 { beta} ^ 2- 2 alpha- 2 beta + 2 alpha beta + gamma = 0 $

But from there, I can not do anything. No suggestion?

Does the integral of the gamma function always occur?

Is the integral of the gamma function already occur in all Department of Mathematics? And I'm not talking about integrals in which you have the product of two gamma functions rotated in opposite directions, or Meijer-G And Fox-H functions & tricks like that – I mean the simple integral of the gamma function. I have never seen it and none seems to mention it.

Abstract algebra – Exact sequence with finite groups: $ 0 to A xrightarrow[]{ alpha} mathbb {Z} ^ d xrightarrow[]{ beta} mathbb {Z} xrightarrow[]{ gamma} B to0 $.

I have this exact sequence of abelian groups:
$$ 0 to A xrightarrow[]{ alpha} mathbb {Z} ^ d xrightarrow[]{ beta} mathbb {Z} xrightarrow[]{ gamma} B to0 $$
with $ A $ and $ B $ finished abelian groups and $ d in mathbb {N} $.

I would like to conclude $ d = $ 1, but I do not know how to use this information $ A, B $ are finished. The only thing I noticed $ gamma $ can not be injective, but I do not know how to proceed. I am a bit rusty in algebra …

Could someone help?

Thank you!

Is there an integrated function for multivariate gamma distribution?

The multivariate gamma distribution <- Wikipedia link with a description of the function.

Is there an integrated function for multivariate gamma distribution?

I discovered that there was a package called GAMMA made with Mathematica 4.0, but it is probably incompatible now …