Skip to main content

Questions tagged [exponential-function]

For question involving exponential functions and questions on exponential growth or decay.

Filter by
Sorted by
Tagged with
-3 votes
0 answers
23 views

Algebra prove with two variables

We have that $(x;y) ∈ ℝ^2$ $ 0\leq x < y$ and $\frac{y}{2024^y} = \frac{x}{2024^x}$ I have to prove that there are infinity $(x;y)$ solutions and there are one $(x;1)$ solution.
Bardocz Roland's user avatar
3 votes
1 answer
90 views

Approximation of $\sigma(n)$ sum.

Investigating: $$\epsilon(n)=\frac{(\pi -3) e^{2 \pi n}}{24 \pi }-\sum _{k=1}^n \sigma(k) e^{2 \pi (n-k)}$$ where $\sigma(n)$ is a divisors sum of $n$. Using long calculations (can not share here ...
Gevorg Hmayakyan's user avatar
0 votes
1 answer
46 views

When is this rational function of exponentials actually rational-valued?

This has come up in my research, and I am sorry if it is obvious. I am looking at the following expression $$ m\frac{\tanh(xm)}{\tanh(x)} = m\frac{e^{2xm}-1}{e^{2xm}+1} \frac{e^{2x}+1}{e^{2x}-1}, $$ ...
Croc2Alpha's user avatar
  • 3,857
12 votes
0 answers
187 views
+50

Inverse function of the Exponential Integral $\mathrm{Ei^{-1}}(x)$

The Exponential integral is defined by $$ \mathrm{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm dt, $$ and has the following expansion $$ \mathrm{Ei}(x) = \log x + \gamma + \sum_{k=1}^\infty \frac{x^...
Nolord's user avatar
  • 398
0 votes
0 answers
31 views

The integral of $\int_0^1 x^a(1+x)^b(1-x)^c e^{-dx^2} dx$

As the title mentioned, I want to get a closed-form result of $\int_0^1 x^a(1+x)^b(1-x)^c e^{-dx^2} dx,$ where $d>0$, $a,b,c$ may be very large numbers. A related integral is $\int_0^1 x^a (1-x)^b ...
jobs adam's user avatar
0 votes
0 answers
20 views

For the complex exponential function, how can we transform the constraints on the exponential term to apply to the whole function? [closed]

Considering $x=e^{jab}$, where $x \in \mathbb{C}$ and $a, b \in \mathbb{R}$, how can I express $a \geq 0$ via $x$ and $b$, without using phase angle function $\angle x$ which will introduce non-...
Haonan Wang's user avatar
0 votes
1 answer
38 views

Solving combination of linear and exponential equation

Trying on my own to prove that $\lim_{n \to \infty}\sqrt[n]n = 1$ have left me with trying to prove that for every $\epsilon \gt 0$, there is $N \in \mathbb N$ such that for all $n \ge N: n \lt (1 + \...
Dark Archon's user avatar
  • 1,553
0 votes
4 answers
93 views

How to prove that: $|e^i|^2=1$

Today my professor wrote the following claim on board: $|e^{(i\pi)/3}|^2=1$ I guess this is true since: $|e^i|^2=1$ But how can I prove the last claim if it's true at all? Side question, is it true as ...
David's user avatar
  • 27
0 votes
1 answer
67 views

Isolating $z$ in the equation $x - 1 = - \frac{1-y^{z+1}-0.5(1-y^z)}{(1-y)y^z}$ [closed]

I have a formula with multiple unknowns: $$x - 1 = - \frac{1-y^{z+1}-0.5(1-y^z)}{(1-y)y^z}$$ The way it is setup now allows to easily calculate $x$, but I would like to reformulate it to isolate $z$, ...
Apo's user avatar
  • 15
-2 votes
0 answers
20 views

How to show there is (no) tie of certain functional form $\frac{x_{1}e^{\mu_{1}}-x_{2}e^{\mu_{2}}}{e^{\mu_{1}}-e^{\mu_{2}}}$. [closed]

I'm trying to prove or disprove the following. For any $\mu_1<\mu_2<\mu_3$ and $x_1<x_2<x_3$, there is no three way ties of $$\frac{x_{1}e^{\mu_{1}}-x_{2}e^{\mu_{2}}}{e^{\mu_{1}}-e^{\mu_{2}...
user23329063's user avatar
0 votes
1 answer
109 views

How to use the Lambert's W function on this exponential equation: $3-x^2=2^x$? [closed]

How to use the Lambert's W function on this exponential equation: $3-x^2=2^x$? I am new to the Lambert's W function and there is barely anything I could find on Google.
Toshiv's user avatar
  • 13
0 votes
1 answer
14 views

I need help filling in some in a step from Fomin's calculus of variations

At the bottom of page 20 from Fomin's book on Calculus of Variations, we have: (1) $\frac{x+A}{c}= \ln( \frac{y + (y^2-c^2)^{1/2}}{c})$ Implies that $y = c \cosh(\frac{x+a}{c})$ Can somebody help me ...
PhysicsIsHard's user avatar
2 votes
0 answers
99 views

Alternative approach for $\int\frac{1}{\tan\left(\frac{1}{2}\csc^{-1} (x)\right) - \tan\left(\frac{1}{2}\sec^{-1}(x)\right)} \, dx$

This is example 2 in my "Integration Using Some Euler-Like Identities" blog post. $$\int\frac{1}{\tan\left(\frac{1}{2}\csc^{-1} (x)\right) - \tan\left(\frac{1}{2}\sec^{-1}(x)\right)} \, dx\...
Emmanuel José García's user avatar
1 vote
1 answer
44 views

$(-a)^x$ versus $-(a^x)$ help

$(-2)^3=-8$ and $(-2)^2=4$, right? And $-(2^3)=-8$ and $-(2^2)=-4$. So that means $(-a)^x$ does not equal $-(a^x)$. My question is why do we never see graphs of $(-a)^x$ then?? I tried graphing $(-2)^...
vergevoyage's user avatar
1 vote
0 answers
31 views

Proving that the limit of $\frac{e^{(tA_\lambda)}u - u}{t}$ as $t\downarrow 0$ exists in a Banach space $X$ for every $u\in X$

Let $X$ be a Banach space and $A$ a closed densely defined (note: not necessarily bounded) linear operator on $X$ and let $\lambda > 0$. Define $$A_\lambda := -\lambda I + \lambda^2 R_\lambda$$ for ...
Epsilon Away's user avatar
0 votes
2 answers
133 views

The fastest solution for $\int \sqrt{x^2+1}\,dx$

The integral $(1)$ is typically attacked using trigonometric substitution, more specifically, using Case II, which involves substituting $x=a\tan{\theta}$. This method leads to having to evaluate the ...
Emmanuel José García's user avatar
5 votes
1 answer
276 views

Question regarding inverse of exponential function

This question is in the context of the following problem Find the inverse of the function $$f:(-\infty, 1] \rightarrow \Biggr[\frac{1}{2}, \infty\Biggr], \text{ where } f(x) = 2^{x(x-2)}$$ I proceeded ...
koiboi's user avatar
  • 734
1 vote
2 answers
115 views

How do we prove that $\lim\limits_{x\to 0} \left(\frac {1+e^{x}}{2}\right) ^{1/x} = \sqrt{e}$?

Wolfram Alpha gives me this solution: $$\lim\limits_{x\to 0} \left(\frac {1+e^{x}}{2}\right) ^{1/x} = \sqrt{e}$$ But I have no idea how to get to that result. I tried using L'Hopital but I found it ...
Atk's user avatar
  • 13
-6 votes
1 answer
34 views

If $\lim_{|s|\to +\infty} \frac{|f(s)|}{e^{s^2}} = 0$ then $\forall m \in \mathbb N$ is it true that ... [closed]

If $\lim_{|s|\to +\infty} \frac{|f(s)|}{e^{s^2}} = 0$, then, for any $m \in \mathbb{N}$ is it true that $\lim_{|s| \to +\infty} \frac{|f(s)|}{|s|^m(e^{s^2} - 1)} = 0 ?$
Lucas Linhares's user avatar
1 vote
1 answer
36 views

$\lim_{x \to 0} \frac{\exp[-x^{-\gamma}]}{x^d}=0$

Trying to show for $\gamma \in (0,1]$ and some natural number $d$ that $$\lim_{x \to 0} \frac{\exp[-x^{-\gamma}]}{x^d}=0$$. I've tried setting $t=x^{-\gamma}$ and you get for the right side limit $$\...
mather's user avatar
  • 33
9 votes
1 answer
157 views

Integral of $\int_{-r}^r e^{\beta y} \arctan\left(\frac{\sqrt{r^2-y^2}}{d}\right) dy$

I'm looking for anything that might help me solve the integral below: $$\int_{-r}^r e^{\beta y} \arctan\left(\frac{\sqrt{r^2-y^2}}{d}\right) dy$$ With $\\{r,d,\beta,y\\}\in\mathbb{R}$, $\beta<0$, $\...
C. Nacke's user avatar
1 vote
0 answers
41 views

Determine if factorial of exponent is bigger than exponent of factorial

For natural values $a$, $b$ determine if $(a^b)! > a^{b!}$. My thoughts : since logarithm is strictly monotonic $a > b \iff \ln(a) > \ln(b)$, let's consider $\ln(a^b)! - \ln(a^{b!})$. Using ...
Vitaliy Volovyk's user avatar
0 votes
0 answers
20 views

Asymptotic Behaviour of $e^{ \alpha \log{( \frac{a}{b})} \log(n)} \times \sum_{t=1}^{n/2} e^ {- \frac{2t}{3} (\log(t) - 3) }$

If $a>b>0$ and $\alpha = \frac{x+1}{2y}$ where $y>0$ and $x\geq 0$. Under which conditions the following summation is asymptotically ($n\to \infty$) upper-bounded by $ k n$ where $k$ is a ...
Jay's user avatar
  • 21
2 votes
0 answers
81 views

How can I prove inequality? [closed]

I have an inequality to prove, $$\left| {M + 2\sum\limits_{m = 1}^M {{{\bar \alpha }_m}} } \right| \le \left| {2M + \sum\limits_{m = 1}^M {{\alpha _m}} } \right|$$ where, $${\alpha _m} = {e^{ - im}}$$ ...
Jay Lee's user avatar
  • 39
0 votes
0 answers
46 views

Periodic Bell Curve?

I'm trying to code a hue selection function in a circular color model (in which each hue is attributed a value between 0 and 1) so I'm working on a Gaussian function and would like it to roundtrip ...
Raphael Jaafari's user avatar
2 votes
4 answers
127 views

Solve the equation $\left(\frac{1+\sqrt{1-x^2}}{2}\right)^{\sqrt{1-x}} = (\sqrt{1-x})^{\sqrt{1-x}+\sqrt{1+x}}$

Solve in $\mathbb{R}$: $ \left(\frac{1+\sqrt{1-x^2}}{2}\right)^{\sqrt{1-x}} = (\sqrt{1-x})^{\sqrt{1-x}+\sqrt{1+x}} $ My approach: Let $a = \sqrt{1-x}$ and $b = \sqrt{1+x}$ so $a^2 + b^2 = 2$. The ...
math.enthusiast9's user avatar
0 votes
0 answers
27 views

Why does the power series form of the exponential equal 0 when evaluated at 0? [duplicate]

Since $e$ is a real number I know that $e^0 = 1$, but when I enter $z=0$ into the power series definition of $e^z$ I get an output of $0$. Am I doing something wrong? $$e^z = \sum_{n=0}^\infty \frac{1}...
guywithllama's user avatar
0 votes
1 answer
51 views

Why is $\sum_{m=0}^{\lfloor xs\rfloor} 2 \binom{s}{m} p^m (1-p)^{s-m} \leq 2\exp{\left(-\frac{2(\lfloor xs\rfloor - sp)^2}{s}\right)}$

I am trying to understand few of the mathematical steps I have encountered in a paper, there are two of them (a) $\sum_{m=0}^{\lfloor xs\rfloor} 2 \binom{s}{m} p^m (1-p)^{s-m} \leq 2\exp{\left(-\frac{...
coolname11's user avatar
1 vote
2 answers
52 views

Properties of sum of two exponential functions

I have some data that can be fit reasonably well with an exponential function. However, a colleague mentioned that it would be better to use the sum of two exponentials: $$ f(x; a, b_1, b_2, \lambda_1,...
jds's user avatar
  • 2,276
-1 votes
1 answer
25 views

express log-log relationship as an exponential relationship [closed]

I have several log(y) ~ a + b * log(x) models that I want to express as exponential relationships. I know this involves an expoential transformation, but how do I solve it? Example: Step 1: log(y) ~ 2 ...
tnt's user avatar
  • 109
0 votes
0 answers
44 views

Integrate a complex exponential

I am trying to calculate this expression: $\int dp \ \exp(-Aip^2+iBp)$ where A and B are real. I integrate p over the reals from -infinity to infinity. How can I visualize this expression and will it ...
ceciled's user avatar
0 votes
1 answer
80 views

Solving $4^x = \log_2(x) + \sqrt{x-1} + 14$ [closed]

Solve in $\mathbb{R}$ the following equation: $4^x = \log_2(x) + \sqrt{x-1} + 14.$ My approach: I noticed that $x=2$ satisfies the equation, then I investigated the intervals $[1,2)$ and $(2,\infty)$, ...
math.enthusiast9's user avatar
0 votes
0 answers
39 views

Writing a twice differentiable function as a linear combionation of exponentials

Let $I$ a non-trivial interval and $f \in D^{2}(I)$ such that $f''(x) = f(x)$ for all $x \in I$. Show that, if there exists some $a \in I$ such that $f(a) = f'(a) = 0$, then $f(x) = 0$ for all $x \in ...
MrGran's user avatar
  • 151
1 vote
1 answer
60 views

Torus and homeomorphism

Let $0<r<a$ be fixed. For $(\alpha,\beta)\in[0,2\pi]^2$ we consider $$ g(e^{i\alpha},e^{i\beta}) = ((a+r\cos\alpha)\cos\beta,(a+r\cos\alpha)\sin\beta,r\sin\alpha) $$ I would like to prove it is ...
coboy's user avatar
  • 1,419
1 vote
1 answer
72 views

Find the number of solutions of $2^x+3^x+4^x-5^x=0$ (without using graphical calculator)

Find the number of solutions of $$2^x+3^x+4^x-5^x=0$$ Answer is given $1$. I tried to take the derivative of the function $$f(x)=2^x+3^x+4^x-5^x\\f'(x)=\ln2 (2^x)+\ln3(3^x)+\ln4(4^x)-\ln5(5^x)$$ but ...
Skdmg's user avatar
  • 872
-1 votes
1 answer
32 views

Sum of Independent exponential random variables 1.0

Help with Exponential Distribution Exercise: Struggling to Eliminate -1 $Let ( X_i \sim \text{Exp}(\theta_1) and\ ( X_j \sim \text{Exp}(\theta_2) ), determinate ( X_i + X_j = U ).$ $\int_0^u \...
Devis's user avatar
  • 29
0 votes
1 answer
25 views

An Inequality with Exponential Terms

I am trying to understand why an inequality in the following post is true: Bounding rows of random matrices. In particular, I am wondering why we have $$ \exp\left(-\frac{t^2/2}{2np+t/3}\right)\le \...
Partial T's user avatar
  • 551
-1 votes
2 answers
50 views

how do you find the initial amount of a decay problem when you dont have one in the problem? [closed]

Suppose a sample of a certain substance decayed to $65.2\%$ of its original amount after $300$ days. What is the half-life (in days) of this substance? (Round your answers to two decimal places.) I ...
The Kold's user avatar
0 votes
0 answers
29 views

A differentiable function that equals a constant times the exponential map

Let $f$ be a differentiable function in $\mathbb{R}$ such that $f'(x) = \alpha f(x)$ for all $x \in \mathbb{R}$, where $\alpha$ is a constant. Prove that $f(x) = f(0)e^{\alpha x}$ for all $x \in \...
MrGran's user avatar
  • 151
0 votes
1 answer
104 views

Why is this continuous function not differentiable?

Let $g : \mathbb{R} \to \mathbb{R}$, such that \begin{equation} g(x)=\begin{cases} 2x + 1 & \text{ if } x < 0,\\ e^x & \text{ otherwise}. \end{cases} \end{equation} I am trying ...
ppch's user avatar
  • 1
0 votes
1 answer
30 views

proof about trigonometrical functions and their relation to the exponential function

Let $w$ be from $\mathbb{C}$ with $|w| = 1$. I need to show that there exists exactly one $t$ from $[0,2\pi)$ such that $w = e^{it}$. My approach would be to rewrite $w$ as $a+bi$ and $e^{it}$ as $\...
maths_xx's user avatar
1 vote
1 answer
108 views

How to compute $\lim_{x\to 0} \frac{e^{ax}-e^{bx}}{x}$?

I'm trying to compute the following limit: $$L=\lim_{x\to 0} \frac{e^{ax}-e^{bx}}{x} \tag{1}$$ And I have to use some of the following limits for it: $$\lim_{x\to 0}(1+x)^{\frac{1}{x}}=e=\lim_{x\to \...
Red Banana's user avatar
0 votes
2 answers
94 views

Proof using the Lambert W function that 1 = 0 - What went wrong?

All values that satisfy $x^2=2^x$ would satisfy $\ln(2)x^3 = x\ln(2)e^{x\ln(2)}$, and would therefore satisfy the relationship $W(\ln(2)x^3) = x\ln(2)$. The problem is that when I graph these ...
Alexandra's user avatar
  • 451
1 vote
0 answers
30 views

$f(x) = \sum_{j=1}^{4}c_{j}e^{r_{j}x} \quad \text{and} \quad g(x) = \sum_{j=1}^{4}d_{j}e^{s_{j}x}$. How to determine $c_{j},d_{j}$?

Let $\alpha_{i},\beta_{i} \in \mathbb{C}$, $\forall i= 1,2$ and $0 < a < b < \infty$. \begin{align} &f^{\prime \prime} + \alpha_{1} f = \alpha_{2} g^{\prime} \quad \text{in} \quad [a,b] \\...
user253963's user avatar
0 votes
0 answers
42 views

Integration Involving Exponential Integral and Gamma Function

I am solving a problem that involves the following integral: $$ \int_x^\infty \frac{e^{-t}}{t^2} dt $$ According to WolframAlpha, for the real part of x being positive, this is equivalent to: $$ \...
S M's user avatar
  • 45
2 votes
1 answer
33 views

A question about the monotonicity of $y_1 = x^e$ and $y_2 = x^{\pi}$

As we know from 1st year Calculus, for function $f(x) = x^n, x \in \mathbb{R}, n \in \mathbb{N}$, if $n$ is odd, the $f(x)$ is increasing; if $n$ is even $f(x)$ is increasing when $x \geq 0$ and ...
ZYX's user avatar
  • 1,164
3 votes
2 answers
111 views

How to prove $e^{-x}-\frac{1}{n}< \left (1-\frac{x}{n} \right )^{n}$?

I was looking the solution given by Simply Beautiful Art of the following problem: https://math.stackexchange.com/a/2029616/952348 In a part of the solution the author claims that How I can prove the ...
HeyHéctor's user avatar
0 votes
1 answer
40 views

basic complex analysis, confusion about complex exponential and its modulus

This question is from the paper 'A new proof of Spitzer's result on the winding of two dimensional Brownian motion' by R Durrett, 1982. Let $D_t = A_t + iB_t$ be a complex Brownian motion. Then $\int \...
patricia's user avatar
0 votes
1 answer
32 views

Exponential of a Multivector in Geometric Algebra: $\exp (xe_1 + ye_2 + be_1\wedge e_2)$

I'm working on understanding the exponential function applied to multivectors in the context of Geometric Algebra, specifically for the multivector $xe_1 + ye_2 + be_1\wedge e_2$. I have found ...
Anon21's user avatar
  • 2,589
8 votes
2 answers
381 views

Maximization of an expression involving exponentials

The problem goes something like this: $$ f(x) = 4^x+8^x+9^x-2^x-6^x-12^x$$ Find the maximum value of f(x). This question appeared on a test in which calculators were not allowed, so please refrain ...
zynox's user avatar
  • 162

1
2 3 4 5
158