I am trying to get the analytical closed form solution for $d^*$ which maximizes an expression $e^{-\lambda d}(z-v-cd)^{\beta} + (1-e^{-\lambda d})(z-cd)^{\beta}$, where $\lambda, z, v, c, \beta$ are constants. Because of the presence of $\beta$, I doubt that analytical closed form solutions exist. However, is it possible to at least get approximate solutions? When I searched for similar questions, I came across some suggestions to use Lambert W function. Is that applicable in this particular context? Any suggestion is appreciated. Thank you.
2026-04-01 07:49:01.1775029741
Closed-form solution/Approximation to Exponential/Power functions
123 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in EXPONENTIATION
- exponential equation with different bases; no logarithms
- Is square root of $y^2$ for every $y>0,y\in\mathbb{R}$?
- Definite sum for $(1+a)^n$
- Fractional exponents definition and the additive law of exponents
- Fourth term in the expansion of $(1-2x)^{3/2}$
- Why is $\int_{0}^{t} e^{nt} \mathrm{\ dt} = \frac{1}{n} \left(e^{nt} - 1\right)$? [solved; notation is also faulty in the first place]
- Exponentiation property of the modulo operator
- When are $\left(\frac{a}{b}\right)^c$ and $\frac{a^c}{b^c}$ equivalent?
- How can I rewrite expression to get log out of exponent
- Compare $2^{2016}$ and $10^{605}$ without a calculator
Related Questions in APPROXIMATION
- Does approximation usually exclude equality?
- Approximate spline equation with Wolfram Mathematica
- Solving Equation with Euler's Number
- Approximate derivative in midpoint rule error with notation of Big O
- An inequality involving $\int_0^{\frac{\pi}{2}}\sqrt{\sin x}\:dx $
- On the rate of convergence of the central limit theorem
- Is there any exponential function that can approximate $\frac{1}{x}$?
- Gamma distribution to normal approximation
- Product and Quotient Rule proof using linearisation
- Best approximation of a function out of a closed subset
Related Questions in CLOSED-FORM
- How can I sum the series $e^{-2}\frac{(3)^n}{n!}\sum_{k=0}^{\infty}\left ( \frac{1}{2}\right )^k\frac{1}{(k-n)!}$
- Computing $\int_0^\pi \frac{dx}{1+a^2\cos^2(x)}$
- Can one solve $ \int_{0}^\infty\frac{\sin(xb)}{x^2+a^2}dx $ using contour integration?
- Finding a closed form for a simple product
- For what value(s) of $a$ does the inequality $\prod_{i=0}^{a}(n-i) \geq a^{a+1}$ hold?
- Convergence of $\ln\frac{x}{\ln\frac{x}{\ln x...}}$
- How can one show that $\int_{0}^{1}{x\ln{x}\ln(1-x^2)\over \sqrt{1-x^2}}\mathrm dx=4-{\pi^2\over 4}-\ln{4}?$
- Exercises about closed form formula of recursive sequence.
- Simplify and determine a closed form for a nested summation
- Direction in closed form of recurrence relation
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?