I am solving a problem where I need to find the charge distribution on a conducting plate, and the field due to it. I’m stuck on this integral.

$$int_0^{infty} frac {x.dx}{(x^2+a^2)^{frac 32}(x^2+b^2)} $$

# Tag: frac

## probability – Calculating the quotient $ frac { mu (E)} { nu (E)} $ of two measures with density w.r.t. the Lebesgue measure

Suppose that $ mu $ and $ nu $ are probability measures on $ mathbb {R} $ with density $ f $ and $ g $ w.r.t. the Lebesgue measure, i.e. $ mu = fdx $ and $ nu = g dx $. Is there an easy way to calculate the qutotient

$$

frac { mu (E)} { nu (E)}

$$

of a measurable set $ E $? This is a quotient of two integrals of $ f $ and $ g $. Can we calculate one single integral over the quotient of $ f $ and $ g $ instead?

## What do I use for $ sum_ {n = 1} ^ { infty} (-1) ^ {n} frac { cos (n)} {n sqrt {n}} $

I think this is a comparison test problem, but I don't know what to compare it with. Any help would be appreciated, thanks. (Attempt to prove convergence).

## Is there a measurable function f: $ mathrm forall a int_ {0} ^ {1} { frac {1} {f (x) -a}} , d mu < infty $?

I want to find a measurable function f: $ mathrm forall a int_ {0} ^ {1} { frac {1} {f (x) -a}} , d mu < infty $, but I can't find such a function. However, I cannot prove that there is no such function. Could someone help me?

## Limits – Prove that $ frac {f (x) – (f * K_t) (x)} {t} bis – Delta f (t bis 0) $ for $ f in C_0 ^ { infty} ( mathbb {R} ^ n) $

Describe

$$ K_t (x) = frac {1} {(4 pi t) ^ { frac {n} {2}}} exp (- frac {| x | ^ 2} {4t}) $$

the heat core, $ f in C_0 ^ { infty} ( mathbb {R} ^ n) $. It is known that $ f * K_t to f (t to 0) $ point by point and $ L ^ p (p ge1) $. How can we prove it? $$ frac {f (x) – (f * K_t) (x)} {t} bis – Delta f (t bis 0) $$

holds ($ Delta $ denotes the Laplace operator)? I got a proof with the spectral resolution, but I wonder if there is a simpler proof with the basic analysis. Thanks a lot.

## ordinary differential equations – what is the singular point of the ode $ frac {dy} {dx} = sqrt {y} + 1 $

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## real analysis – why $ sum_ {n = 1} ^ infty frac {4 sin pi n} { pi pi n ^ 2} sin nx = 2 sin (x) $?

I used Wolfram Alpha to calculate this integral

$$ frac {1} { pi} int _ {- pi} ^ { pi} 2 sin x sin nx ; dx = frac {4 sin pi n} { pi pi n ^ 2} $$

So if $ n in mathbb {Z} $ That answer must be $ 0 $ Law?

Then I used Wolfram Alpha again for the calculation

$$ sum_ {n = 1} ^ infty frac {4 sin pi n} { pi pi n ^ 2} sin nx $$

and it is called

$$ sum_ {n = 1} ^ infty frac {4 sin pi n} { pi pi n ^ 2} sin nx = 2 sin (x) $$

How is that possible?

## $ Frac {1} {6-n ^ 3} $ applies to all $ n> 1 $. Is it convergent or divergent?

So we're using the threshold comparison test, and we've got that $ lim_n rightarrow infty frac {n} {6-n ^ 3} $. I know it's all over $ sum $ from $ n = 1 $ to $ infty $ from $ a_b $, but I do not know what $ a_b $ is.

## Algebra precalculation – Find $ n $ such that $ 365 left (1 – ( frac {364} {365}) ^ n – n frac {364 ^ {n-1}} {365 ^ n} right)> $ 1

i have to find $ n $ so that $ 365 left (1 – ( frac {364} {365}) ^ n – n frac {364 ^ {n-1}} {365 ^ n} right)> 1 $. The answer is $ n ge 28 $. I understand expanding the equation, rearranging it, taking the logarithm

$$ log (364)> log (365) + n log (364/365) + log (n) + (n-1) log (364/365). $$

I don't know what to do next. I would be happy if you could give me a hint.

## nt.number theory – is there a non-negative sequence $ a_p $ so that $ sum_p frac {a_p} {p} $ converges but $ sum_p frac { sqrt {a_p}} {p} $ diverges?

Is there a real, non-negative sequence? $ a_p $ indexed on the prime numbers so that $ sum_p frac {a_p} {p} $ converges however $ sum_p frac { sqrt {a_p}} {p} $ diverged? If so, what is an example of such a sequence, and if not, how can this be proven?

(This was the result of examining the presumption distance for multiplicative functions in analytical number theory. A sequence that meets these conditions is required to find a multiplicative function $ f $ so that $ sum_p frac {1 – Re (f (p))} {p} $ but converges $ sum_p frac {| 1 – f (p) |} {p} $ diverges.)