Saturday, September 25, 2021

Recent Questions - Mathematics Stack Exchange

Recent Questions - Mathematics Stack Exchange

Is the cantor function a fractal projected in 1D?

Posted: 25 Sep 2021 08:23 PM PDT

Projective geometry projects objects with one dimension, into another dimension.

Is in that sense the cantor fractal projected into 1D to make the cantor function?

Is there a theory for projecting fractals into integer dimensions as a generalization of projective geometry?

Cantor function

A function f is defined by , find the implicit domain of f (please)

Posted: 25 Sep 2021 08:06 PM PDT

A function f is defined by

enter image description here

find the (implicit) domain of f

Distribution of inverse matrix entries from a matrix with entries distributed iid U(0,1)

Posted: 25 Sep 2021 08:05 PM PDT

I had an idea for a "small" math problem which I thought would be fairly reasonable to solve but it doesn't seem to be. I have formalized it below. Not a lot of success with it, however.

Let $\mathbf{M}$ represent a square matrix of dimension $n$ and let its entries $m_{j,k}$, be independent and identically uniformly distributed random variates between 0 and 1. What is the distribution of the entries of the matrix $\mathbf{W}$ if $\mathbf{W}$ is the inverse of $\mathbf{M}$, or $\mathbf{W} = \mathbf{M}^{-1}$ and as $n$ approaches infinity?

My initial approach was to expand the size of the square matrix and augmenting it with additional value which are also random variates which are also iid distributed U(0,1). If $n$ is sufficiently large then this expansion should have a diminishing impact on the distribution.

Looking forward to any inputs you have!

Regarding the equivalence of two inequalities concerning Schwarz Lemma

Posted: 25 Sep 2021 08:04 PM PDT

I am trying to study the Nevanlinna pick Problem. I wanted to know how the two given inequalities are equivalententer image description here

How to know the positivity of a given polynomial

Posted: 25 Sep 2021 08:02 PM PDT

Let $f(x)=\dfrac{1}{(2p+1)(2p+2)}x^{2p}+\sum\limits_{i=1}^{2p}\binom{2p+1}{i}\dfrac{1}{i+1}x^{i-1}.$ How do you know about the positivity of the $f(x)$? I am trying to induction on $p$ as well as combine the term $1,x, x^2$ and $1,x^3, x^6$, etc as well as combine the terms $1,x,x^2$ and $x^2, x^3, x^4$, etc to get the sum of squares. However, it is hard to control when the coefficient depends on the binomial with $p$. I also tried to use Sturm theorem as well as Eneström's theorem to determine its zeros points and from there to see the positivity but I am not successful either. It would be great if someone can give me some help. Thanks.

Number Fields (Marcus) page 16 Some Applications

Posted: 25 Sep 2021 08:02 PM PDT

In Some Applications Marcus is trying to justify that the units in $\mathbb{A} \cap k$ are the elements having norm $\pm 1$.

I understand the concepts behind the justification, that if you had an algebraic element that was a unit then it should have norm $\pm 1$ and that if you have an algebraic element that has norm $\pm 1$ then its inverse is also an algebraic element:

Using Corollary 2 (If $\alpha$ is an algebraic integer, then $T(\alpha)$ and $N(\alpha)$ are integers) above and the fact that the norm is multiplicative, it is easy to see that every unit has norm $\pm 1$

Here's what I tried: If $\alpha$ is an algebraic integer and a unit (there exists $\beta \in \mathbb{A} \cap K$ s.t. $\alpha \cdot \beta = \beta \cdot \alpha =1$), then by Corollary 2, $N(\alpha), N(\beta)$ are integers, and $N(\alpha\cdot\beta) = N(\alpha)N(\beta) = N(1) = 1$ since $\mathbb{Q}$ is fixed pointwise. The only integer divisors of $1$ are $\pm 1$, $N(\alpha) =\pm 1$.

Not sure where to start on how $N(\alpha) = \pm 1$ implies $\frac{1}{\alpha} \in \mathbb{A}$?

On the other hand if $\alpha$ is an algebraic integer having norm $\pm 1 $ then Theorem 4 shows that $\frac{1}{\alpha}$ is also an algebraic integer (since all conjugates of $\alpha$ are algebraic integers).

Theorem 4.

$$T(\alpha) = \frac{n}{d}t(\alpha)$$ $$N(\alpha) = (n(\alpha))^\frac{n}{d}$$

where $n = [K:\mathbb{Q}]$.

Here $\alpha$ has degree $d$ over $\mathbb{Q}$ and $t(\alpha), n(\alpha)$ denote the the sum and product of the $d$ conjugates of $\alpha$ over $\mathbb{Q}$.

Theorem 26.10 (the Cartier Equality) in Matsumura's Commutative Ring Theory

Posted: 25 Sep 2021 08:01 PM PDT

I am trying to understand the proof of Theorem 26.10 in Matsumura's Commutative Ring Theory, and I cannot understand the last step.

To expand a bit, let $k\to A\to B$ be ring homomorphisms, then we have a natural map $\Omega_{A/k}\otimes_A B\to \Omega_{B/k}$ where $\Omega$ is the module of relative differentials. Let $\Gamma_{B/A/k}$ be the kernel of this map, the so-called imperfection module. The theorem then states:

Let $k$ be a perfect field, $K$ an extension of $k$, and $L$ a finitely generated extension field of $K$. Then $$\mathrm{rk}_L\Omega_{L/K} = \mathrm{tr.deg}_KL + \mathrm{rk}_L\Gamma_{L/K/k}$$ where $\mathrm{rk}_L$ is the dimension of a vector space over $L$ and $\mathrm{tr.deg}_K$ is the transcendence degree of a field extension.

Following the proof we can reduce to the case where $L = K(\alpha)$ where char $K = p$, and $\alpha^p = a\in K$, but $\alpha\notin K$. He then says that writing $L = K[X]/(X^p-a)$ that we can see

\begin{align*} \Omega_{L/k} & = (\Omega_{K[X]/k}\otimes L)/Ld\alpha\\\\ & = (\Omega_{K/k}/Kda)\otimes L \oplus Ld\alpha \end{align*}

and $d\alpha\neq 0$. Furthermore, since $k$ is a perfect field, we have $a\notin K^p = kK^p$, so that in $\Omega_{K/k}(= \Omega_{K/\Pi}$ where $\Pi$ is the prime subfield) we have $da\neq 0,\mathrm{rk}\Omega_{L/K} = 1$ and $\mathrm{rk}\Gamma_{L/K/k} = 1$, so that the equality holds.

I fail to follow anything starting from the calculation of $\Omega_{L/k}$, and have exhausted all of my ideas which did not lead to anything useful.

Construct an arrangement of numbers that the absolute value of differences between adjacent numbers are all different

Posted: 25 Sep 2021 08:01 PM PDT

In the original duscussion: Different Differences between an Arrangement of Numbers, the problem has the limitation: 3 is in the third place, and it provide the answer choices from $12$ to $16$. I want to know if there is a way to construct all possible arrangement without the limitation, or at least calculate how many ways there are?

My attempt:
I consider arranging $1,2,3,4,5$ to let the sum of any consecutive numbers is not $0$ mod $6$, and I got $(4,1,3,5,2)$ and $(1,4,3,2,5)$ (reverse is eligible too). Let starting number from $1$ to $6$, and add the adding sequence to see if it works.

For ex: use the sequence $(4,1,3,5,2)$ and mod $6$
$1$ $5$ $6$ $3$ $2$ $4$ $-$ X
$2$ $6$ $1$ $4$ $3$ $5$ $-$ O
$3$ $1$ $2$ $5$ $4$ $6$ $-$ X
starting from $4$,$5$,$6$ is the reverse of upper sequence.

use the sequence $(1,4,3,2,5)$ and mod $6$
$1$ $2$ $6$ $3$ $5$ $4$ $-$ X
$2$ $3$ $1$ $4$ $6$ $5$ $-$ X
$3$ $4$ $2$ $5$ $1$ $6$ $-$ O
starting from $4$,$5$,$6$ is the reverse of upper sequence.

Other sequences from $(2,5,3,1,4)$ and $(5,2,3,4,1)$ are:
$2$ $4$ $3$ $6$ $1$ $5$
$1$ $6$ $2$ $5$ $3$ $4$

However, there are many other sequences that don't meet the condition I set. For ex:
$4$ $2$ $3$ $6$ $1$ $5$ is the sequence from $4$ and the difference mod $6$ $(4,1,3,1,4)$, because the first $4$ is actually $-2$ and the second $1$ is actually $-5$.

I am a little bit confused about the process of construction. Are there appropriate thinking process to deal with this problem? Thank you very much.

Variation and Probability - Chances of possible outcomes

Posted: 25 Sep 2021 07:58 PM PDT

I really can't wrap my head around this one. I was looking at crypto mass NFT image creations and saw some are randomly generated avatars. These avatars have multiple different "things" that can appear and I was curious how you would work out the "rarity" of a certain creation.

For example an avatar may be a suit and a base ball cap (a top and a hat) however there are multiple different accessory types that could be added such as:

Hat Jewellery Clothes Shoes Props Background

Each of these types are filled with certain different assets such as hats may include:

Baseball cap - 5% Helmet - 2% Scuba goggles - 20% Police Hat - 10% Nothing - 63%

Each one of these has a certain chance of appearing.

I believe this means that if I for example got an avatar that had nothing but a baseball cap it doesn't mean the chance of me getting that is 5% but actually less due to the other types that could have been added.

My question is, if I had access to all % chances of all assets within each type how would I work out the chances of my avatar being "created" if it had something like:

Baseball cap Necklace Suit No props No Background

A sufficient condition for being of the second category

Posted: 25 Sep 2021 07:51 PM PDT

A set $U$ of a metric space $X$ is said to be of the second category if $U$ is not a countable union of nowhere dense sets. Here is a sufficient condition for a subset to be of the second category from the textbook that I am trying to prove.

Proposition: $U$ is of the second category in $X$ if, whenever $U \subset \bigcup_n F_n$ with each $F_n$ close, at least one $F_n$ must have nonempty interior.

Please check my solution for correctness. Assume on the contrary that each $\text{Int} \ F_n =\emptyset$. Since $F_n$ is closed, each $F_n$ is nowhere dense. We can write $U$ as $U = \bigcup_n (U \cap F_n)$. If we can show that each $U \cap F_n$ is nowhere dense, then we are done (by contradiction). Since $(U \cap F_n) \subset F_n$, we have $\overline{U \cap F_n} \subset \overline{F_n}$. And so, $\text{Int }\overline{U \cap F_n} \subset \text{Int } \overline{F_n} = \emptyset$, and we have the desired contradiction.

How to solve for $y$ here?

Posted: 25 Sep 2021 07:52 PM PDT

I've been working on an initial value problem that I want to solve for $y$ in. I've reached: $$\frac{y-4}{y+1}=(e^{-t}+C)^5$$ But I'm not sure where to go from here. How can I solve for $y$? Thanks.

Note: if you all need the steps I took to get to this point, I can provide that, but either way I don't know how to solve for $y$ from here.

Independent random sample problem

Posted: 25 Sep 2021 07:44 PM PDT

Let $\{X_i\}$ be a random sample, where each $X_i$ is independent and distributed according to the following:

$$p(x)= \begin{cases} 0.5, if -1 \leq x\leq 1\\ 0, otherwise \end{cases}$$

(i) Is $X_i$ normally distributed? Explain.

(ii) What is the mean and variance of $X_i$?

(iii) Suppose we let $Y_i = X_i - X_{i - 1}$. Compute $E[Y_i]$ and $Var(Y_i)$.

(iv) Would your answers to (iii) above change if $X_i$ and $X_{i - 1}$ were not independent? Justify.

My attempt:

(i) Not sure. Can anybody explain?

(ii) If $-1\leq x\leq 1$, then $E[X_i]= \int_{-1}^{1} 0.5x \,dx = 0$, $E[X_i]= \int_{-1}^{1} 0.5x^2 \,dx = \frac{1}{3}$ and so $Var(X) = \frac{1}{3}$. Otherwise, $E[X_i] = 0$ and $Var(X) = 0$.

(iii) We have $E[Y_i] = E[X_i - X_{i - 1}] = E[X_i] - E[X_{i - 1}] = E[X_i] - E[X_i] = 0$ and $Var[Y_i] = Var(X_i - X_{i - 1}) = Var(X_i) + Var(X_{i - 1}) = Var(X_i) + Var(X_i) = 2Var(X_i)= \begin{cases} \frac{2}{3}, if -1 \leq x\leq 1\\ 0, otherwise \end{cases}$

(iv) If $X_i$ and $X_i$ are not independent, then the expansion formula for $E[X_i]$ stays the same however formula for $Var(X_i)$ changes because the covariance becomes possibly non-zero. That is, $E[X_i - X_{i - 1}] = E[X_i] - E[X_{i - 1}]$ regardless of whether $X_i$ and $X_{i - 1}$ are independent but $Var(X_i - X_{i - 1}) = Var(X_i) + Var(X_{i - 1}) - 2cov(X_i, X_{i - 1})$ instead. Hence only the variance may change.

Is this correct? I am especially not sure about $(i)$ so any assistance is much appreciated.

Evaluate $\int_0^1 \ln(-\ln x)\frac{x^{a}}{\sqrt{-\ln x}}dx$

Posted: 25 Sep 2021 07:42 PM PDT

I am having trouble with the integral:

Evaluate $$\int_0^1 \ln(-\ln x)\frac{x^{a}}{\sqrt{-\ln x}}dx$$

My Attempt

let $u=-\ln(x)\rightarrow x=e^{-u}\rightarrow dx=-e^{-u}du$ bounds: $(0,1)\rightarrow (\infty,0)$ $$ \int_0^1 \ln(-\ln x)\frac{x^{a}}{\sqrt{-\ln x}}dx=\int_{\infty}^0 \ln(u)\frac{e^{-au}}{\sqrt{u}}(-e^{-u}du)=\int^{\infty}_0 \ln(u)\frac{e^{-(a+1)u}}{\sqrt{u}}du $$ let $(a+1)u=w \rightarrow du=\frac{dw}{a+1}$

$$ \int^{\infty}_0 \ln(u)\frac{e^{-(a+1)u}}{\sqrt{u}}du=\int^{\infty}_0 \ln(w/(a+1))\frac{e^{-w}}{\sqrt{w/(a+1)}}(\frac{dw}{a+1})=\frac{1}{\sqrt{a+1}}\int^{\infty}_0 \ln(w/(a+1))\frac{e^{-w}}{\sqrt{w}}dw $$ $$ \frac{1}{\sqrt{a+1}}\int^{\infty}_0 \ln(w))\frac{e^{-w}}{\sqrt{w}}dw-\frac{1}{\sqrt{a+1}}\int^{\infty}_0 \ln(a+1))\frac{e^{-w}}{\sqrt{w}}dw=\frac{1}{\sqrt{a+1}}\int^{\infty}_0 \ln(w))\frac{e^{-w}}{\sqrt{w}}dw-\frac{\ln(a+1)\sqrt{\pi}}{2\sqrt{a+1}} $$

How do you solve from here? Have I used the correct method? thank you for your time

g : Z × Z → Z and g(m, n) = m^2 − n ^2 Is this proof valid

Posted: 25 Sep 2021 08:05 PM PDT

Determine whether or not g is surjective Would this be a valid proof?

If we take any y, and set m=0. Then we have the equation y = 0 - n^2 and solve for n.

Then we'll have the square root of y equal n. If y = 2 then n = the square root of 2 which does not belong to the codomain therefore the function is not surjective.

If $\lim_{x \to a} f(x)$ exist, then $\lim_{x \to a} g(x)$ DNE implies $\lim_{x \to a} [f(x) + g(x)]$ DNE

Posted: 25 Sep 2021 08:07 PM PDT

I need some help with this proof question that I am finding it hard to show. I am uncertain if this method is the correct way of showing. Here is the problem:

If $\displaystyle \lim_{x \to a} f(x)$ exists, show that $\displaystyle \lim_{x \to a} g(x)$ DNE implies $\displaystyle \lim_{x \to a} [f(x) + g(x)]$ DNE

I first wrote this statement as $p \to (q \to r)$ and I intend to show this by contradiction using the $\epsilon-\delta$ definition.

So we assume $\displaystyle \lim_{x \to a} f(x)$ exists, meaning that $\forall \epsilon > 0 \exists \delta_1 > 0$ such that $0 < |x - a| < \delta_1 \to |f(x) - L| < \epsilon$.

Also assume (for the sake of contradiction) that $\displaystyle \lim_{x \to a} g(x)$ DNE (meaning $\exists \epsilon > 0 \forall \delta_2 > 0$ such that $0 < |x - a| < \delta_2$ implies $|g(x) - M| \geq \epsilon$) but $\displaystyle \lim_{x \to a} [f(x) + g(x)]$ exists (meaning $\forall \epsilon > 0 \exists \delta = \min(\delta_1, \delta_2) > 0$ such that $0 < |x -a | < \delta$ implies $|(f + g)(x) - (L + M)| < \epsilon$.

I know that we can write the function $g(x) = [f(x) + g(x)] - f(x)$, so substituting this into the definition for $\displaystyle \lim_{x \to a} g(x)$ we have \begin{align*} 0 < |x - a| < \delta_2 &\to |g(x) - M| \\ &=\left|\left[[f(x) + g(x)] - f(x)\right] - [[L + M] - L]\right| \geq \epsilon \end{align*} Which should be a contradiction since if $f(x) + g(x)$ corresponds to the limit $L + M$ which exists, by assumption, and then $f(x)$ corresponds to the limit $L$ which also exists, we then have two limits that actually exists, therefore this implies that $\displaystyle \lim_{x \to a} g(x)$ must also exist. Therefore, proving the statement.

I'd appreciate some advice or any corrections that should be corrected with this proof.

Temperature inside a hemisphere

Posted: 25 Sep 2021 08:18 PM PDT

The surface of a hemisphere is held at temperature $T_0$ and its base is at $0$. Calculate the temperature throughout the hemisphere.

Tangent space of fiber bundle (or vector bundle)

Posted: 25 Sep 2021 08:27 PM PDT

The Connection (vector_bundle) article of Wikipedia mentions about Ehresmann connection that a smooth map $s:M\to E$ ($E$ is any vector bundle over $M$) has a differential $ds:TM\to TE$, this means there is tangent spaces on the vector bundle $E$.

What is the condition that tangent space can be defined on fiber bundle? Article about fiber bundle and vector bundle doesn't mention that the bundle is also a smooth manifold.

IMO if both the base space and the fiber space are smooth manifolds then the fiber bundle is also a smooth manifold whose tangent space at any point has a dimmension of the sum of the dimmensions of the both spaces? In the above special case, the fiber of $E$ is $R^n$ which is a smooth manifold.

Is that right?

Evaluate $\int_0^\infty \frac{e^{-x}\sin(x)}{\sqrt[3]{x} } dx$

Posted: 25 Sep 2021 07:51 PM PDT

I am having trouble with the following integral

Evaluate $$\int_0^\infty \frac{e^{-x}\sin(x)}{\sqrt[3]{x} } dx$$

$$\int_0^\infty \frac{e^{-x}\sin(x)}{\sqrt[3]{x} } dx=\int_0^\infty 3ue^{-u^3}\sin(u^3)du$$ let $u=\sqrt[3]{x}\rightarrow dx=3x^{2/3}du$

How does one proceed from here? Thank you for your time.

How do you find the value of $\lim\limits_{x \to \infty} \frac{3+2x^2}{x-1}+\frac{ 5x^2+x+2x^2} {1-x} $? [closed]

Posted: 25 Sep 2021 07:44 PM PDT

How do you find value of this limit , note that the power in numerator is higher than the power of denominator ,2 in the numerator and 1 in denominator, so I can't solve it dividing by the highest power in the denominator, to summarize I don't know the way to solve this problem which is to find the value of this limit $\lim\limits_{x \to \infty} \dfrac{3+2x^2}{x-1}+\dfrac{ 5x^2+x+2x^2} {1-x} $?

Is there any layman proof or explanation for area as an definite integral?

Posted: 25 Sep 2021 07:39 PM PDT

I am reading a chapter on integrals from a School textbook. In the chapter, a function called the area function is defined based on the definite integral. It is as follows

$$A(x) = \int\limits_{a}^{x} f(x) dx$$

The following are two theorems are given in the textbook

Theorem 1

Let $f$ be a continuous function on the closed interval $[a, b]$ and let $A(x)$ be the area function. Then $A'(x) = f(x)$ for all $x \in [a, b]$

Theorem 2

Let $f$ be a continuous function on the closed interval $[a, b]$ and $F$ be an anti-derivative function of $f$. Then $\int\limits_{a}^{b} f(x) dx = [F(x)]_{a}^{b} = F(b)-F(a)$.

but it is mentioned that the proof of the theorems stated is beyond the scope of the book I am studying.

Is there any elementary or layman proof/explanation for understanding how the definite integral quantifies the area under the curve?

Other Two Points of Rectangle Giving Two Points and Width

Posted: 25 Sep 2021 08:15 PM PDT

I need to find the other two points of a rectangle given two points shown in black in the attached drawing. The missing points are showed in yellow width is in red which I have as well. The rectangle can be arbitrarily rotated.

I also know if the points are on the same side or on opposite sides of the rectangle. So basically I know which of the four configurations the system is in before having to solve for it.

There are four possible arrangements each of which I need to solve for as shown.

rectangles image

Partial differential equations related to diffusion

Posted: 25 Sep 2021 08:06 PM PDT

Setting: I am studying solutions of the partial differential equation

$w_t - cw_y-D(w_{xx}+w_{yy})=0 $,

where $c$ and $D$ are real constants. The domain of the solutions shall be $(x,y)\in (0,L) \times { (0,L)}$ and the initial condition is $w(x,y,0)=f(x,y)$. I furthermore demand Dirichlet boundary conditions $w(0,y)=w(L,y)=w(x,0)=0$ and $w(x,L)=u_o=const$.

Let $w(x,y,t)=u(x,y,t)+v(x,y)$ be a solution of (1).

Question: Under which conditions can I find a function $v(x,y)$ such that $u(x,y,t)$ satisfies homogeneous boundary conditions?

Are the left and right unitor maps of the unit object in a monoidal category the same?

Posted: 25 Sep 2021 07:37 PM PDT

Suppose we have a monoidal category $\mathbb{C}$, with monoidal product $\otimes$, monoidal unit $I$, left unitor components $I \otimes A \overset{\lambda_A}{\rightarrow} A$ and right unitor components $A \otimes I \overset{\rho_A}{\rightarrow} A$. In this case is it true that $\lambda_I = \rho_I$? If so what is the proof?

Can we retrieve $X$ from its odd moments?

Posted: 25 Sep 2021 08:21 PM PDT

The moment problem asks whether there exists a random variable $X$ with given moments. One way to do this: if $X$'s MGF converges about a neighborhood of $0$, we know the MGF uniquely characterizes $X$ in which case the pdf can be retrieved by applying the inverse laplace transform to its MGF.

My question is, if the odd moments are such that $\sum_{k\geq0} \frac{\mathbb{E}X^{(2k+1)k}}{k!} s^k$ converges around $0$, is knowing the odd moments of $X$ sufficient to uniquely characterize/retrieve $X$?

Edit: PhoemueX's comment notes if we take all odd moments zero, then the inversion is non-unique. But I don't see how to show non-uniqueness for any other circumstance or know if this is a degenercy.

Motivation: Consider if $X$ is a non-negative random variable such that the MGF of $X$ exists and converges about an open interval containing $0$. Then its possible to retrieve $X$ from moments $\mathbb{E}X^{nk}, k\in \mathbb{N}$, for fixed $n$. This is since the MGF $\mathbb{E}e^{X^n t}$ converges too, hence uniquely characterizes $X^n$, but since $X$ was positive this gives $X$. (Note if $n$ is even, the positive restriction was necessary).

Evaluation of ${\sum\limits_{\Bbb N} (\text {ker}(x)+\text{kei(x)})=\sum\limits_1^\infty \text K_0\left(\sqrt ix\right)= 0.133691… - 0.7256312… i}$?

Posted: 25 Sep 2021 08:25 PM PDT

$\large \text{Introduction:}$

This summation is related to this other Bessel Summation question:

$$\mathrm{\sum\limits_{-\infty}^\infty Ai(x)=1}\ \text{and}\ \sum\limits_{-\infty}^0 \mathrm{Bi(x)}$$

$\large \text{Goal Sum:}$

The actual question uses the Modified Bessel Function of the Second Kind and corresponding Kelvin functions The sum may have a slightly different value. It would probably be easiest to use integral representations as alternative forms of the functions may not work well. I am looking for 2 solutions: one closed form for the kei(x) sum and one closed form for the ker(x) sum. This is because all other summations of just a function ended up with a closed form. Also try contour representations of which I am unfamiliar:

$${\sum_{\Bbb N} \text{ker}(x)+i\text{kei}(x)= \sum_1^\infty \text K_0\left(\sqrt ix\right)= \sum_{x=1}^\infty \int_0^\infty \frac{\cos\left(\sqrt{i}tx\right)}{\sqrt{t^2+1}}dt=\sum_{x=1}^\infty\int_0^\infty \cos\left(\sqrt ix\sinh(t)\right)dt=0.133691752819604391549325780771600891… - 0.725631207729182631737443031218031025… i}$$

Here is a summand plot. Note the sum starts for $x\ge 1$:

enter image description here

$\large \text{Abel-Plana Integral Representation:}$

Another method is to use the Abel-Plana formula which does work as shown here with the difference term evaluated. The simpler integral over the summand only has a complicated closed form. Here the integral of the summand in the last step can be found by evaluating the closed form antiderivative from $[-1,\infty]-[-1,1]$:

$${\sum_0^\infty \text{ker}(x+1)+i\ \text{kei}(x+1) =\frac{\text{ker}(1)+i \ \text{kei}(1)}{2}+\int_0^\infty \text {ker}(x+1)+i\ \text{kei}(x+1) dx+\int_0^\infty \frac{i\ \text{ker}(1-ix)-\ \text{kei}(1-ix)-i\ \text{ker}(1+ix)+\ \text{kei}(1+ix)}{e^{2\pi x}-1}dx=\frac12 \text K_0\left(\sqrt i\right)+\int_0^\infty \text K_0\left(\sqrt i(x+1) \right)dx +i\int_0^\infty \frac{\text K_0\left(\sqrt i(1-ix) \right)-\text K_0\left(\sqrt i(1+ix) \right)}{e^{2\pi x}-1}dx}$$

$\large \text{Other Integral Representations:}$

Using @Jack Barber's Floor function integral solution in the following question gives this result. Please see the bolded "Kelvin functions" link for more Generalized Kelvin function information:

Evaluation of $$\sum_{x=0}^\infty \text{erfc}(x)$$

$$\sum_\Bbb N(\text{ker(x)}+ i\text{kei}(x))=\sum_\Bbb N \text K_0\left(\sqrt i x\right) =\sqrt[4]{-1}\int_0^\infty\lfloor x\rfloor \text{kei}_1(x)dx+\sqrt[-4]{-1}\int_1^\infty \lfloor x\rfloor \text{ker}_1(x)dx=(-1)^{-\frac34}\int_1^\infty \lfloor x\rfloor \text K_1\left(\sqrt[4]{-1} x\right)dx$$

Here is the integrand plot:

enter image description here

Let's now use the Fractional Part/Sawtoothwave function. Note that the point discontinuities can be ignored as a result of the integral operator:

$$\text{frac}(x)=\mod{(x,1)}=\text{sawtoothwave}(x)=\boxed{\{x\}}=x-\lfloor x\rfloor\implies x-\{x\}=\lfloor x\rfloor$$

Therefore our goal sum can be expressed as the following. This is the closed form integral of $x\text{kei}_1(x),x\text{ker}_1(x)$. The integral on $[1,\infty]$ is the same as on $[0,\infty]-[0,1]$, but the integral of the $x\text{ker}_1(x)$ function integral on $[0,\infty]$ is just $-\frac\pi 2$ while for $x\text{ker}_1(x)$, it is just $0$ on the same interval meaning that:

$$\sum_1^\infty (\text{ker}(x)+i\text{kei}(x))= \sqrt[4]{-1}\int_0^\infty (x-\{x\}) \text{kei}_1(x)dx+\sqrt[-4]{-1}\int_1^\infty (x-\{x\}) \text{ker}_1(x)dx = (-1)^{-\frac34}\left(\frac\pi 2+\int_0^1 x \text{kei}_1(x)dx+\int_1^\infty\{x\} \text{kei}_1(x)dx \right)+ (-1)^{\frac 34}\left(\int_0^1 x \text{ker}_1(x)dx+\int_1^\infty\{x\} \text{ker}_1(x)dx \right) $$

$$\sum\limits_1^\infty \text K_0\left(\sqrt ix\right) = (-1)^{-\frac34}\int_1^\infty (x-\{x\})\text K_1\left(-\sqrt[4]{-1} x\right)dx =\sqrt[4]{-1}\left(\frac{i\pi}2+\int_0^1x \text K_1\left(\sqrt[4]{-1} x\right)+\int_1^\infty\{x\}\text K_1\left(\sqrt[4]{-1} x\right)dx\right)$$

There are many alternate forms for the floor function, so many more integral representations are possible in terms of related functions.

Here is a plot of that Fractional Part function times the Bessel-Kelvin type function integrand:

enter image description here$\large \text{Conclusion:}$

Note that I could have made a typo. An exact form answer is needed and a closed form is optional. This question was found as the summand graph decreases rapidly and after a lot of terms, and converges quickly to over $50$ digits in just the first $300$ terms. We now have $2$ working integral representations of the constant, so how can we evaluate them or the sum representations? Please correct me and give me feedback!

Absolute values and Quadratic

Posted: 25 Sep 2021 08:08 PM PDT

The difference of the roots of the quadratic equation $x^2 + bx + c = 0$ is $|b - 2c|$. If $c \neq 0$, then find $c$ in terms of $b$.

I know Vieta, for sum and product of roots of a quadratic, but am not seeing how to apply any of those tools to find the absolute difference $|b-2c|.$ I'm definitely missing a nice trick to do this. Solutions?

How to find values such that the curves $y=\frac{a}{x-1}$ and $y=x^2-2x+1$ intersect at right angles?

Posted: 25 Sep 2021 08:18 PM PDT

Problem:
Find all values of $a$ such that the curves $y = \frac{a}{x-1}$ and $y = x^2-2x+1$ intersect at right angles.

My attempt:
First, I set the two curves equal to each other:
$ \frac{a}{x-1} = x^2 - 2x + 1 $
$ \frac{a}{x-1} = (x-1)^2 $
$ \frac{a}{x-1}(x-1) = (x-1)^2(x-1) $
$ a = (x-1)^3 $
$ \sqrt[3] a = x-1 $
$ \sqrt[3] a + 1 = x $

​ I found the derivative of the first curve:
$ y = \frac{a}{x-1} $
$ y = a(x-1)^{-1} $
$ y' = -a(x-1)^{-2} $
$ y' = \frac{-a}{(x-1)^{2}} $

Then I found the derivative of the second curve:
$ y = x^2 - 2x + 1 $
$ y' = 2x - 2 $

Next, I multiplied them together and set them equal to -1:
$ \frac{-a}{(x-1)^{2}} ⋅ 2x - 2 = -1 $

$ \frac{-2ax + 2a}{(x-1)^2} = -1 $

$ \frac{-2a(x-1)}{(x-1)^2} = -1 $

$ \frac{-2a}{x-1} = -1 $

$ \frac{-2a}{x-1} ⋅ (x-1) = -1(x-1) $

$ -2a = -x+1 $

$ a = \frac{-x+1}{-2} $

Now I am unsure how to finish the problem. Do I substitute what I found for $x$ into $ a = \frac{-x+1}{-2} $?
But my problem shows that there are two possible answers for $a$ so I am confused.

Any help would be appreciated!
Thank you in advance!

Find the coordinates of the point a line meets a plane

Posted: 25 Sep 2021 08:07 PM PDT

The line through the point (1, 2, 6) orthogonal to the plane $x + 3y + z = 4$ meets the plane at the point X. Find the coordinates of X.

I've attempted to illustrate what I've interpreted as what's going on in the following graphic: enter image description here

The normal line (1, 3, 1) indicates where in space the plane is. The point (1, 2, 6) is a point on that normal line. Using the vector equation of a line, we can construct a way to express the coordinates of r, in the equation:

$$ r = a + t(n)$$

In which $a$ is the vector with (1, 2, 6) and $n$ is the vector with (1, 3, 1).

With this in mind, we can find the coordinates using the Cartesian equation of a line

$r_x = a_x + t(n_x)$

$r_y = a_y + t(n_y)$

$r_z = a_z + t(n_z)$

We obviously need to find the scalar here. According to lecture note solutions, we need to apply $r_x$, $r_y$, and $r_z$ coordinates into the plane equation. My main queries are as follows:

  • What purpose does the vector equation of a plane serve? In terms of its derivation, it's based on the statement $n·(r - r_1)$ = 0, but what better does this do to describe the plane than simply the coordinates of the normal line? I know that the normal line isn't a function, so that isn't very useful for us to use, what information does this tell us?
  • Are any of my assumptions, or anything about my graphic wrong? If so, why? Finally, why do we plug the values into the plane equation? Please take me through the logical forethought of that.
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