## integration – Integral of \$intlimits_0^{2pi } {{e^{acos (theta – b) + ccos (d – theta )}}dtheta } \$?

I know that the integral of

$$intlimits_0^{2pi } {{e^{acos (theta – b))}}dtheta } = 2pi {I_0}(a)$$

Where, $${I_0}(a)$$ is the Modified bessel function of the first kind.

I am trying to find the integral of $$intlimits_0^{2pi } {{e^{acos (theta – b) + ccos (d – theta )}}dtheta }$$. Can I transform this integral into Bessel function or some known function?

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## binance – Sent old ERC-20 THETA tokens to mainnet wallets

I recently sent my old Theta token to Binance as well as the Theta Wallet online (wallet.thetatoken.org), and have yet to see a balance in either wallet despite confirmations. I realize now that I sent the old tokens that hadn’t been swapped to mainnet wallets. Is there any hope of recovering my old tokens and swapping them for mainnet THETA and TFUEL?

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## Does it make sense to say Big Theta of 1? Or should we just use Big O?

Says say $$f (x) = Theta (1)$$ Provide additional information when you say $$f (x) = O (1)$$?

Intuitively, nothing grows more slowly than a constant, so there should be no additional information in this case if Big Theta is given over Big O.

## Theta functions – Is \$ sum_ {n in mathbb {Z}} e ^ {- (n- mu) ^ 2/2 sigma ^ 2} le sum_ {n in mathbb {Z} } e ^ {- n ^ 2/2 sigma ^ 2} \$ for all \$ mu \$ and all \$ sigma \$?

I looked at discrete Gaussian distributions and came to the following assumption. I would be very happy to have proof (or proof).

Guess. To let $$mu in (0.1)$$ and $$sigma ^ 2> 0$$. Then $$sum_ {n in mathbb {Z}} e ^ {- (n- mu) ^ 2/2 sigma ^ 2} le sum_ {n in mathbb {Z}} e ^ { -n ^ 2/2 sigma ^ 2}.$$

Numerical evidence supports this assumption.

In the continuous analogue (i.e. replace $$sum _ { mathbb {Z}}$$ With $$int _ { mathbb {R}}$$), that's an equality. Indeed as $$sigma ^ 2 to 0$$the discrete sum becomes a closer approximation to the continuous integral and the inequality almost becomes an equality. (Even for $$sigma ^ 2 = 1$$is the difference only $$2.7 times 10 ^ {- 8}$$.)

On the other hand if we take $$sigma ^ 2 to infty$$then the infinite sum is dominated by a single term. The inequality becomes approximate $$forall mu ~~~~~ e ^ {- mu ^ 2/2 sigma ^ 2} le e ^ 0,$$ that is trivially true.

So we see it in both extremes ($$sigma ^ 2 to 0$$ and $$sigma ^ 2 to infty$$) the presumption applies. This is further proof that it applies to all values ​​of $$sigma ^ 2$$.

These sums can be expressed in Jacobi theta functions. However, I don't see how helpful this is.

## Why \$ ( pi + theta) \$ is not in the interval \$[pi,0]\$?

To let $$theta in ( pi, 0)$$

A complex number (Z) is:

$$Z = r (-cos theta – isin theta)$$

Why $$theta$$ cannot be the same $$( pi + theta)$$ but can be the same $$(- pi + theta)$$ then?

Aren't they the same thing?

## Algorithms – What is the big theta of \$ ( log n) ^ 2 + 2n + 4n + log n + 50 \$?

$$f (n) = ( log n) ^ 2 + 2n + 4n + log n + 50$$

I am trying to prove mathematically that $$f (n)$$ falls under the temporal complexity of $$theta (( log n) ^ 2)$$.

I have to come to the end $$f (n) leq C ( log n) ^ 2$$for a positive constant $$C$$ and $$x geq k$$.

What I tried is:

$$( log n) ^ 2 + log n leq 2 ( log n) ^ 2$$

i want to add $$6n + 50$$ to both sides, but cannot find the constant $$c$$ by algebra in $$c ( log n) ^ 2$$.

I was just trying to set random values ​​from $$c$$ close $$c ( log n) ^ 2 geq 6n$$ true and $$c = 100$$ works for $$n$$ larger than a value, but there is a mathematical way to find that value from $$c$$ and therefore find the big theta of this function?

## partial derivative – get \$ z = f (x, y) \$ with \$ x = rcos theta \$ and \$ y = rsen theta \$ prove it

To get $$z = f (x, y)$$ With $$x = rcos theta$$ and $$y = rsen theta$$
Prove that
$$frac { partially ^ 2 z} { partially r ^ 2} = frac { partially ^ 2 z} { partially x ^ 2} cos ^ 2 theta + 2 frac { partially ^ 2 z } { partial x partial y} sen theta cos theta + frac { partial ^ 2 z} { partial y ^ 2} sen ^ 2 theta$$

Find the solution, but I don't know if it's right

Find $$frac { partial z} { partial r} = frac { partial z} { partial x} cos theta + frac { partial z} { partial y} sen theta$$

If this is the only thing it does, ^ 2 to get the result. Is this procedure okay?

## Asymptotics – Prove that for all functions g: N -> R> = 0 and all numbers a in R> = 0, if g in omega (1), then a + g in theta (g)

Here is a more readable version of the question:

Prove that for all functions $$g: mathbb {N} to mathbb {R} ^ { geq 0}$$and all the numbers $$a in mathbb {R} ^ { geq 0}$$, if $$g in Omega (1)$$ then $$a + g in Theta (g)$$

What I've done so far:

In order to $$a + g in Theta (g)$$. $$a + g in mathcal {O} (g) Keil a + g in Omega (g)$$,

If we expand our assumption, we get:

$$exists c_1, n_1 in mathbb {R} ^ {+}, forall n in mathbb {N}, n geq n_1 implies g (n) geq c_1$$

To prove $$a + g in Omega (g)$$::

If we expand the definition, we get:

$$exists c_2, n_2 in mathbb {R} ^ {+}, forall n in mathbb {N}, n geq n_2 implies a + g (n) geq c_2 * g (n)$$

To let $$c_1 = 1$$ and $$n_2 = n_1$$, To let $$n in mathbb {N}$$, Accept $$n geq n_2$$, To prove $$a + g (n) geq c_2 * g (n)$$,

$$g (n) = g (n) \ implies g (n) geq g (n) \ Leftrightarrow c_1 * g (n) geq c_1 * g (n) \ Leftrightarrow g (n) geq g (n) text {(since c_1 = 1 )} \ Left right arrow a + g (n) geq g (n) text {(enlargement of the left side since a in mathbb {R} ^ { geq 0} )}$$

To prove $$a + g in mathcal {O} (g)$$::

If we expand the definition, we get:

$$exists c_3, n_3 in mathbb {R} ^ +, forall n in mathbb {N}, n geq n_3 implies a + g (n) leq c_3 * g (n)$$

I'm fighting here because I'm not really sure what its value is $$c_3$$ I should use or how to derive one. (I tried to use my assumption of $$g (n) geq c_1$$ but I don't really know where to go from there). Any help is greatly appreciated and I apologize for any formatting errors in advance. Thank you very much.

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## Number theory – perfect square problem \$ a ^ 2 + b ^ 2 – 2ab \$ cos (\$ theta \$)

I'm trying to find out if $$a ^ 2 + b ^ 2 – 2ab$$cos ($$theta$$) can always be a perfect square if $$a, b$$ are clearly positive integers> $$1$$With $$a$$ < $$b$$and cos ($$theta$$) $$epsilon$$ )$$0, a / 2b$$(. All values ​​of cos ($$theta$$) in this interval must be irrational or rational numbers $$p / q$$ and if they're rational, it seems to me $$q$$ could be a factor of $$from$$. $$a$$, or $$b$$what the expression would do $$a ^ 2 + b ^ 2 – 2ab$$cos ($$theta$$) an integer. I've been kicking this around for some time and I can't see a way into the problem. Suggestions would be very welcome.

## Graphic – Why does Schlick's approximation contain a \$ (1- cos theta) ^ 5 \$ term?

The approximation writes the reflection coefficient as$$R (θ) = R_0 + (1-R_0) (1-cosθ) ^ 5, R_0 = left ( frac {n_1-n_2} {n_1 + n_2} right) ^ 2.$$Why is the exponent 5? Schlick 1994 leads this exponent in Eq. (24) with the claim that it is the correct Fresnel approximation but without explanation.

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