Questions tagged [exponential-function]

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

955 questions
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Exponential function expansion

Can someone please explain why this is true? $$\exp\left(\ln(x) + \frac{\ln(x)}{x} + \frac1x + O(\frac{1}{x^2})\right) = x \left(1 + \frac{\ln(x)}{x} + \frac{1}{x} + O(\frac{\ln(x)^2}{x^2})\right)$$...
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Integral form of a limit

Consider: $$\lim_{n \rightarrow\infty} \sum_{k=0}^n {n\choose k} \frac{1}{n^k}$$ One might consider using the binomial expansion to obtain: $$\lim_{n \rightarrow\infty}\left(1+\frac{1}{n}\right)^n$$ ...
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Applications of the logarithm property $(\log_pq)(\log_rs)=(\log_rq)(\log_ps)$

I recently came across this intriguing property of logarithms: $(\log_pq)(\log_rs)=(\log_rq)(\log_ps)$, which is linked to a result known as the "chain rule" for logarithms. This seems to be less well-...
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Logarithmic Paper

enter image description here I have a question related to logarithmic paper. Does the midpoint shows 0.5*(10^(-1))? Or less than that because of increase of logarithmic also? I am curios about the ...
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Exponential family of distributions

A family of pdfs or pmfs is called an exponential family if $$f(x|\theta) = h(x)c(\theta) \exp \left(\sum_{i=1}^{k} w_{i}(\theta) t_{i}(x) \right)$$ so, I wonder if this distribution belongs to the ...
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Integral of third order polynomial exponential

I am looking for approximated or exact solution of \begin{align} I = \int_R \exp(cx^3-ax^2+bx)dx \end{align} where $a,b,c$ are complex numbers defined as: \begin{align} c &= \frac{1}{3}i\pi\phi'''...
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Formal Power Series Composition with Exponential

I have seen formal power series expressed as $$B(z) = \sum_{i=1}^{\infty} b_i{z}^i,$$ but then also as $$B(z) = \sum_{i=1}^{\infty} \frac{b_i}{i!}{z}^i.$$ Is there a significant difference between ...
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Newtons Law Of Cooling (And Heating)

Rule is: $D= A.e^{-kt}$, Where: $k,a$ are elements of real numbers, $D$ is the difference between the temperature of the item and the surrounding air, and t is the time in hours since the object ...
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Financial Mathematics--Finding Compounding Period given Annual and Effective Interest Rates

I'm trying to find a compounding period C when given an annual interest rate r and effective annual yield i. I'm working with the following equation: $i=(1+r/C)^C-1$ I'm having trouble re-writing ...
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Question about the connection between exponential and logarithmic functions

Does this make sense to anyone? What advice would you give me to clarify my reasoning and explanation? One of the really "neat" features of the exponential function: $$f(x)=e^x$$ is the fact that the ...
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What is different between $\frac{1}{1+\lambda x}$ and $\exp{(-\lambda x)}$

I want to choose a function $f(x)$ which has properties: $f(x)$ closes to $0$ when $x$ goes to $+\infty$ . I have two option for that $f(x)=\frac{1}{1+\lambda x}$, where $\lambda$ is tuning ...
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Confidence interval for exponential - is it the shortest possible?

The confidence interval for an exponential distribution is said to be: $$\frac{2n\bar{x}}{\chi^2_{1-\alpha /2,2n}}<\frac{1}{\lambda}<\frac{2n\bar{x}}{\chi^2_{\alpha /2,2n}}$$ In general we aim ...
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Solve for deceleration in exponential decay equation

$$y = y_0 + v_0\cdot d\frac{1 - d^{t}}{1 - d}$$ $y$ = final position $y0$ = initial position $v0$ = initial velocity $d$ = deceleration $t$ = elapsed time How do I solve for $d$? Specifically, ...
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Taylor of $\ln(f(exp(x))))$?

Let $f(x) = \sum a_n x^n$ Such that The $a_n$ are real and $f(a),f ' (a) , f " (a) > 0$ for any real $a > 0$. Let $\ln(f(exp(x))) = \sum b_n x^n$. Let $c_n = a_n - b_n$. For a given $f$ ...
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exponential generating function for the number of ways to arrange marbles in a line

Say we have red, green, and blue marbles that we are arranging in a line of length n. We need to use an even number of blue marbles, at least two red marbles, and at most two green marbles. I am ...
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Three almost-integers of the form $ce^{H_a+H_b}\approx 2^k\pm1$

The approximation $$H_n\approx log(2n+1)$$ https://math.stackexchange.com/a/1602945/134791 suggests that the harmonic number for composite odd numbers might be close to the sum of the harmonic ...
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How can I apply Rouche's Theorem here?

How many solutions lie in the left half-plane? $$f(z) = z^3+2z^2-z-2+e^z=0$$ My work so far: Factoring the polynomial, moving the exponential term over to the RHS, and taking the modulus of both ...
I need to prove the following integral relation: $$\lim_{T \to (1-i \epsilon) \infty} \mathrm{\int_{-T}^T dt_1 dt_2 e^{-ia |t_1|} \cdot e^{-ib |t_2|} \cdot e^{-ic |t_1-t_2|}} = \frac{-2}{(a+b)(b+c)} + ... 0answers 14 views exponentialfunction poofs with generating functions I have two things to proof. (e^{ax})^{-1}=e^{-ax} and (e^{ax})^{m}=e^{(ma)x} I know the power series of e^x and e^{ax} and that e^{ax}e^{bx}=e^{(a+b)x} I tried it forward and backward: (... 0answers 74 views Compute \int_0^1 \frac{\exp(u^3 - 4/3 u) du }{\sqrt {1 - u^2}}  Compute the integral$$\int_0^1 \frac{\exp(u^3 - 4/3 u) du }{\sqrt {1 - u^2}} $$I assume contour integration is the easier way ? 0answers 65 views Memoryless property implies E(\max\{X_1,X_2,\ldots,X_n\})- E(\min\{X_1,X_2,\ldots,X_n\})=E(\max\{X_1,X_2,\ldots,X_{n-1}\}) X_1,X_2,\ldots,X_n \sim \mathrm{Exp}(\lambda), \quad \text{i.i.d.} How to show the following equation by using memoryless property:$$E(\max\{X_1,X_2,\ldots,X_n\}) - E(\min\{X_1 , X_2, \ldots, X_n\...
Fenton and Griffiths (2008) in Risk Assessment in Geotechnical Engineering give the following representation of a stochastic process: Z(x_j) = \sum_{k=0}^{N} A_k \cos(\omega_k x_j) + B_k \sin(\...