Integral with exponential function

Could you please help me in solving the following integration. Thank you in advance.

Integrate(((Sqrt(g*(x + h)/y))^V) * Exp(-u * Sqrt(g*(x + h)/y)/n) * Exp(-L*x), {x, 0, Infinity})

enter image description here

linear algebra – Computing the exponential of a $2 times 2$ matrix using trace $0$ matrices

It is an easily proved fact that for a $2times 2$ traceless matrix $A$,

$$ e^A = cosleft(sqrt{det(A)}right)I + frac{sinleft(sqrt{det(A)}right)}{sqrt{det(A)}}A$$

Problem 2.7 of Lie Groups, Lie Algebras, and Representations by Bryan Hall asks to use this fact to compute $exp(X)$, where

$$ X = begin{pmatrix}
4 & 3\
-1 & 2
end{pmatrix}$$

In other words, I have to write $X$ in terms of traceless matrices, and employ the above fact. My question is: is there a systematic way to do this?

My idea to solve this problem is to write $X = X_1 + X_2$, where $X_1$ is traceless, $X_2$ is diagonal or nilpotent, and $(X_1, X_2) = 0$, and compute the exponent using $e^{X_1 + X_2} = e^{X_1}e^{X_2}$. For example, I tried the most obvious thing:
$$X = begin{pmatrix} -2 & 3\-1 & 2end{pmatrix} + begin{pmatrix} 6 & 0\0 & 0end{pmatrix},$$

but the two matrices above do not commute.

plotting – Find exponential function that becomes constant at 1

samplelist={{0., 15}, {0.031746, 14}, {0.0634921, 13}, {0.0952381, 
  12}, {0.126984, 11}, {0.15873, 11}, {0.190476, 10}, {0.222222, 
  9}, {0.253968, 9}, {0.285714, 8}, {0.31746, 8}, {0.349206, 
  7}, {0.380952, 7}, {0.412698, 7}, {0.444444, 6}, {0.47619, 
  6}, {0.507937, 6}, {0.539683, 5}, {0.571429, 5}, {0.603175, 
  5}, {0.634921, 4}, {0.666667, 4}, {0.698413, 4}, {0.730159, 
  4}, {0.761905, 4}, {0.793651, 3}, {0.825397, 3}, {0.857143, 
  3}, {0.888889, 3}, {0.920635, 3}, {0.952381, 3}, {0.984127, 
  2}, {1.01587, 2}, {1.04762, 2}, {1.07937, 2}, {1.11111, 
  2}, {1.14286, 2}, {1.1746, 2}, {1.20635, 2}, {1.2381, 2}, {1.26984, 
  2}, {1.30159, 2}, {1.33333, 1}, {1.36508, 1}, {1.39683, 
  1}, {1.42857, 1}, {1.46032, 1}, {1.49206, 1}, {1.52381, 
  1}, {1.55556, 1}, {1.5873, 1}, {1.61905, 1}, {1.65079, 1}, {1.68254,
   1}, {1.71429, 1}, {1.74603, 1}, {1.77778, 1}, {1.80952, 
  1}, {1.84127, 1}, {1.87302, 1}, {1.90476, 1}, {1.93651, 
  1}, {1.96825, 1}, {2., 1}}
(!(Plot of sample list and exponential function)(1))(1)


Show(Plot(10 Exp(-t /(0.5)), {t, 0, 2}), Plot(1, {t, 0, 2}))

(!(Plots of both functions together)(1))(1)

I need an exponentially decaying function that fits my data and becomes constant (equal to 1 here) ultimately.

Asymptotic solution involving the exponential integral.

I’m currently currently reading through a paper where the function

$$R(t) = 1 -sqrt{dfrac{2t}{beta}} – dfrac{1}{beta} displaystyle{left(dfrac{2t}{3} + Vsum_{n=1}^inftyleft(dfrac{2e^{-n^2pi^2kt}}{n^2pi^2} – frac{text{erf}(pi nsqrt{kt})}{n^3pi^{5/2}sqrt{kt}} right) right)} tag 1$$ ,

is given, and fast convergence for the involved infinite series (in the case of $t ll 1$) is illustrated by using the identity:

$$displaystyle{sum_{n=1}^infty e^{-n^2pi^2kt} = -frac{1}{2} + frac{1}{2sqrt{pi kt}}}left(1 + 2 sum_{n=1}^infty e^{-n^2/kt}right) tag 2$$

to derive

$$R(t) sim 1 -sqrt{dfrac{2t}{beta}} – dfrac{1}{beta} left(dfrac{2}{3}(1+kV)t – dfrac{Vsqrt{kt}}{sqrtpi} – displaystyle{frac{2V}{sqrtpi}sum_{n=1}^inftyleft( sqrt{kt}e^{-n^2/kt} – frac{n^2}{sqrt{kt}}E_1left(frac{n^2}{kt}right) right) }right)$$

as $beta to infty$, where $E_1$ is the exponential integral defined by

$$displaystyle{E_1(z) = int_{z}^infty frac{e^{-xi}}{xi}dxi}$$.

I’ve tried to derive this asymptotic myself, however I have made little progress. It seems the writers may have modified (1) before substituting in (2) multiplied with $kt$, however I have no idea what this possible modification of (1) could be. I would be greatly appreciative if someone could give an outline of its derivation.

algebra precalculus – Can I flip the exponential growth function to get a logarithmic growth function

High school textbook exponential growth:

$$A(t)=A^{kt}$$

Where $A$ is the initial amount, $t$ is the number of time periods and $k$ is the growth rate. With this formula I can determine the function for a given variable using two known points. Then, once I have the function I can estimate the amount after $t$ time periods.

I wondered if there’s an equivalent logarithmic function? I’m working with a variable that appears to follow log growth and I would like to know if there’s a function I can use, similar to exponential growth, where I can plug in an initial amount $A$ and estimate the future amount $A(t)$ after t time periods. I would like to know if I can estimate $k$ in the same way as with a exponential model?

I was thinking I could try to flip a exponential line to get it logarithmic.

E.g. just playing around with the exponential function on Desmos, I can swap signs and add fixed amounts to get a line resembling a looking closer to a logarithmic function, e.g. $y=-10^{-left(0.5cdot t-1right)}+10$

Looks like this:
enter image description here

Is it possible to wrangle the exponential growth formula to make it work for logarithmic growth where I can estimate future values of $Y$ after $t$ time periods?

operating systems – Confusion about the definition of “Exponential Average”

I was having doubt regarding the concept of “exponential averaging”. Till now I have come across the concept twice and they seem to be quite similar (except that they have minor but sharp differences, which sort of confuses me). Once in Operating System and another in Computer Networks.


Galvin

The above picture is from Galvin OS text, which associates $alpha$ (smoothening factor with the actual value , here actual CPU burst time).


Forouzan

The above picture is from Forouzan Computer Network text where the $alpha$ is associated with the estimated value (of RTT).


Kurose and Ross

The above picture is from Kurose and Ross Computer Networks text and it associates $alpha$ with the actual value (of RTT) ( which is similar to what done in Galvin)

Is there any standard definition of this concept? It kind of seems confusing to me.

simplifying expressions – Collect Exponential With Pattern

I have some enormous expression with many complex exponentials. Some of these are time varying, others are just phase terms. Here is a small piece of the expression that I hope fully captures the behavior of the full expression for the purposes of this question.

expr = 4 E^((6 I L w)/c + (4 I L wm)/c + 2 I t wm) E0^2 r1^2 + 
 4 E^((4 I L w)/c + (4 I L wm)/c - I t wm) E0^2 r1 r2 tr^2 + 
 4 E^((8 I L w)/c + (2 I L wm)/c + I t wm) E0^2 r1 r2 tr^2 + 
 4 E^((8 I L w)/c + (4 I L wm)/c + 2 I t wm) E0^2 r1 r2 tr^2

I would like to collect these expressions so that each oscillating frequency is factored out like so:

E^(-I t wm) (4 E^((8 I L w)/c + (2 I L wm)/c) E0^2 r1 r2 tr^2 + 
    4 E^((4 I L w)/c + (4 I L wm)/c) E0^2 r1 r2 tr^2) + 
 E^(2 I t wm) (4 E^((6 I L w)/c + (4 I L wm)/c) E0^2 r1^2 + 
    4 E^((8 I L w)/c + (4 I L wm)/c) E0^2 r1 r2 tr^2)

I can do this with Collect(expr, {E^(2 I t wm), E^(- I t wm)}) just fine, but I have a lot of different frequencies and I don’t want to manually enumerate them. I guess I could try to extract them from the big expression, but I feel like I should be able to do this with just Collect. I’ve tried the most obvious thing Collect(expr, E^(__ t)). But that doesn’t seem to work.

Problem of Analysis about Matrix Exponential, Infimum and Limit

Let $f: mathbb{R}^{n^2} times mathbb{Z}^{n} longrightarrow mathbb{R}$ defined by
$$
f(X,z) = prod_{i=1}^{n} |x_i z|,
$$

where $x_i$ is the $i$-th row of $X$ and $x_iz$ is a dot product of $x_i$ and $z$.
My question is:
Is it true that
$$
inf_{z neq 0} lim_{X rightarrow 0} f(e^{X+C}, z) = lim_{X rightarrow 0} inf_{z neq 0} f(e^{X+C}, z) ?
$$

Some observations that I deduce about this:

  1. $e^{X} = displaystylesum_{k=0}^{infty} dfrac{X^{k}}{k!}$ is infinitely differentiable and therefore $e^{X+C}$ is continuous.
  2. I think that you don´t need to know about the constant $C$, but $C = log b$ and the definition is that $e^{C}=b$. $C$ is a matrix, i.e., $C in mathbb{R}^{n^2}$.
  3. Actually, still I believe that we don´t need to know about the function $f$ the same way that I wrote above. I think that the problem can be formulated as follows:

Given $f: mathbb{R}^{n^2} times mathbb{Z}^{n} longrightarrow mathbb{R}$ continuous, is it true that $inf_{z neq 0} lim_{X rightarrow 0} f(X, z) = lim_{X rightarrow 0} inf_{z neq 0} f(X, z)$?

So I can´t do how prove it. I tested with some examples in Matlab and Wolfram Mathematica and the equality has always been valid.

One of the ways which I thought was to “open” the expressions of both sides of equality and try to reach the same result. To reach the infimum, I thought that $z$ should have all coordinates equal to zero, except 1 coordinate. In this case, I have $z neq 0$ and maybe this is a vector that works to prove it. But I didn´t get anywhere.

calculus – Deriving logistic growth equation from the exponential

I’m following along with this: https://jdyeakel.github.io/teaching/ecology/section9/ and I just want to make sure my derivation is correct since this website seems to use prime notation not for derivatives, but for different values. (And that really took me an embarrassingly long time to figure out)

So, it’s as if we start off with exponential growth $frac{dN}{dt}=kN$ and then, for small population $N$, $k=b_0 – d_0$ (where those $0$‘s are the initial values, or y-intercepts).

So the equation becomes $frac{dN}{dt}=(b_0 – d_0)N$ but then, as population increases, we don’t want constant values, but linear equations $b$ and $d$. And these linear equations are are $b_0 -aN$ and $d_0 + cN$

Now we’d get $frac{dN}{dt}=((b_0 -aN) – (d_0 + cN))N$

So I guess my question is “is this how you make the connection between exponential growth and logistic growth”? Do we start with one and then build the other one on top of it, or should it be derived in a completely different way. In other words:

  • Do we start with exponential and then convert it into logistic

  • Do we start with constants for birth and death rate and then make them into lines

  • Do we start with the assumption things aren’t density-dependent, and then make it so that it is

nt.number theory – Can inverse of a non-logarithmic function be an exponential function?

Given a function $f(x)$ for which $f(x)$ is not a logarithmic function, can the its inverse $f^{-1}(x)$ be an expoenetial function?

In particular, I am considering functions of the form
$$
f(x) = frac{x}{(a + x)^{epsilon}}
$$

for which the domain is $mathbb{Z}^+$, $a in mathbb{Z}^+$, and $epsilon in (0, 1)$.

My guess is that it is possible that $f^{-1}(x)$ is exponential. But I cannot find a valid example (or a counterexample).