All Questions
Tagged with calculus-and-analysis simplifying-expressions
193 questions
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Efficient way to calculate the limit of a very long expression [closed]
I want to calculate the limit of a very long expression, but simplifying the expression using Simplify or FullSimplify doesn't ...
1
vote
1
answer
153
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How to assume "much greater than" when there are multiple assumptions? [closed]
Assume $r_{o1} \gg R_S, r_{o2} \gg R_S, g_{m1} r_{o1} \gg 1, g_{m2} r_{o2} \gg 1$.
$$Z_{out} = \frac{r_{o1} (R_S + r_{o2})} {r_{o2} + R_S (1- g_{m1} g_{m2} r_{o1} r_{o2})}$$
Here is what I tried to ...
- 1,892
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2
answers
131
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Issue with variable within Sum[] bounds
I have the following sum that starts from a variable lower bound index $n$:
$$\sum _{y=n}^x \frac{n^2! \left(n^2\right)^{-x} \mathcal{S}_x^{(y)}}{\left(n^2-y\right)!}$$
I tried to compute a closed ...
- 197
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0
answers
135
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Comparing simplification result by Mathematica with Maple [closed]
I use both Maple and Mathematica. Was porting small code from Maple to Mathematica.
In one small computation I noticed this very strange difference and is really serious difference which can affect ...
- 151k
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44
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3
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2
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312
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Problem computing a limit
I have the following function in Mathematica:
$\frac{x! \sum _{i=1}^j (-i+j+1) S_{j+1}^{(-i+j+2)} x^{-i+j+1}}{(x-j)! \left((1-x)_j\right){}^2}$
Defined as:
...
- 197
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1
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146
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NIntegrate can't solve this complicated multiple integration with absolute value in Exp and Cos functions
I'm trying to use NIntegrate to calculate this complicated multiple integration, which has absolute value in the Exp and Cos functions.
$$
\int_{-\infty}^{\infty} dx_1 \int_{-\infty}^{\infty} dx_2 \...
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1
answer
90
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Separation of variables
Is it possible to replicate the same procedure of placing the functions that depend on $r$ on one side of the equation and on the other side leave the equations that depend on $\theta$ in Mathematica? ...
- 169
1
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1
answer
47
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Issue with FullSimplify in StirlingS2 definition
I have previously asked a question about a problem simplifying an expression in Mathematica. The solution was to expand the definition of StirlingS2.
This time, I ...
- 197
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1
answer
49
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Problem calculating mean of probability distribution
I have a discrete probability distribution where $P(X=x)=(Factorial[n]/(n^x))*StirlingS2[x-1,n-1]$. So, when trying to calculate its mean with the following property: $\mu = \sum_{i} x_i \cdot P(x_i)$,...
- 197
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3
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156
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Finding $\frac{\cos(b)}{\cos(a)}$ given $\cos(x) e^{i y} = \frac{\sin(a)}{\sin(a+b)}$ and $-i \sin(x) e^{i y} = \frac{\sin(b)}{\sin(a+b)}$
I have real $x$ and $y$, and I have the following implicit definitions of (complex) $a$ and $b$:
$$\cos(x) e^{i y} = \frac{\sin(a)}{\sin(a+b)}$$
$$-i \sin(x) e^{i y} = \frac{\sin(b)}{\sin(a+b)}$$
I ...
- 247
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0
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57
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Speeding up the output of DSolveChangeVariables
I am trying to change the variable in a differential equation using the above mentioned function. This is what I did:
...
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3
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1
answer
179
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2
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1
answer
126
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How to enforce a range for variable when taking a limit?
I am trying to compute the limit of the following complex expression with respect to $\lambda$. It gives me an error which I think has to do with defining the range of the variable. In my case it is ...
- 315
2
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2
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258
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Taking the derivative of Re[f]
I have looked at similar posts, such as this one, but none of the solutions have worked for me. I have an imaginary function, and I have tried to obtain its real part using ...
- 21
4
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1
answer
143
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Strange result simplifying higher order BesselJ [duplicate]
Consider the following integral:
$$\int_0^1 Z_n^m(r)\ J_m(\rho r) r dr$$.
The solution of this should contain a single Bessel function:
$$(-1)^{(m-n)/2}\ J_{n+1} (\rho)/\rho$$ (see https://www.osti....
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77
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Limit giving indeterminate result
I have a function $r_h(v)$ given by,
$$r_h (v) = \left(\frac{m_0}{2 g} e^{-g v_f} \left(-1+e^{-g(v - v_f)} \right) \right)^{1/5}$$
where $m_0$ and $g$ are just numbers. I want to take the limits of ...
- 725
2
votes
2
answers
130
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How to prevent evaluation without simplification?
I am trying to compute a sequence of terms, which have a form similar to $g(z)$ at different points $z$. For $z$ close to 0, I have observed that the terms are evaluated as 0/0 (indeterminate form) ...
- 121
8
votes
3
answers
3k
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Why doesn't Mathematica recognise two integrals as equal?
The following expression
$$\int_{0}^{t} f(t_0) \,dt_0 - \int_{0}^{t} f(t_1) \,dt_1 = 0$$
is zero because it is just a change of integration variable.
Why doesn't Mathematica give zero in this case?
<...
- 310
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2
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191
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0
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0
answers
57
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Manipulating Dirichlet characters and L functions
I read some basic knowledge about characters and L functions, and would like to play around with them in MMA.
I tried to do the following things, but ending in minor success. (MMA notation used)
...
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1
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4
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186
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Evaluation of an integral using Mathematica or otherwise
I need to find a closed form (in terms of known functions) using Mathematica or otherwise of $$\Re\left(\int_{\frac{1}{2}}^{1}\frac{\tan^{-1}\left(\frac
{1-x}{\sqrt{-i-x^2}}\right)}{\sqrt{-i-x^2}}\ dx\...
- 301
4
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1
answer
69
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Confusing cases in Piecewise
The following
FullSimplify[Integrate[Exp[I ω t] Sin[ω x]/ω, {ω, 0, ∞},
Assumptions -> t > 0 && x > 0]]
gives a piecewise output of the form
I ...
- 63
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2
answers
476
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Calculating a limit using Wolfram cloud or otherwise
Using Wolfram cloud or otherwise, I need to show $$\lim_{n\to\infty} \frac{n\ 4^{2n}}{e^{2n}}\ \left\{\binom{2n}{n} d_{2n}a_n\right\}\leq \frac{3}{4}$$ where $$a_n:= -\sum_{j=0}^n\binom nj^2(2(H_{n-j}-...
- 301
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161
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Closed form for a sum involving Bernoulli numbers
I need a closed form for the sum $$\sum_{n=1}^{\infty}\frac{(-1)^{n-1}(2^{2n}-1)\pi^{2n}B_{2n} {2n+4 \choose m}x^{2n+4}}{(2n)!}$$
where $0<x<1$, $B_{2n}$ denotes Bernoulli numbers , $m\in\mathbb{...
- 301
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vote
1
answer
84
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How to collect products of DiracDelta functions?
After taking a FourierTransform, I have the following expression containing multidimensional products of delta functions (what I am actually working with is far longer than this excerpt):
...
- 113
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3
answers
332
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How to prove an identity involving a hypergeometric function?
How to prove with the help of Mathematica the following statement?
$$ {}_2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};-1\right)=\frac{\pi -3 \sqrt{3} \log \left(\sqrt[3]{2}-1\right)-6 \tan^{-1}\left(\...
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468
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Evaluating ${}_5F_4\left(1,1,1,1,1;\frac{3}{2},2,2,2;-\frac{1}{4} \right)$
Using Mathematica, how can I find a closed-form expression (in terms of elementary functions) of $$ {}_5F_4\left ( 1,1,1,1,1;\frac{3}{2},2,2,2;-\frac{1}{4} \right ),$$ where ${}_5F_4$ represents the ...
- 301
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0
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31
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What is preventing Mathematica from simplifying $-\frac{1}{3}\frac{d}{dx}\ln|1+x^{-3}|$ even when assuming $x\in\mathbb{R}$? [duplicate]
For the input
D[-1/3*Log[Abs[1 + x^(-3)]], x]
the output is
Derivative[1][Abs][1 + 1/x^3]/(x^4*Abs[1 + 1/x^3])
so I put
...
- 157
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1
answer
155
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Evaluation of integral with parameters
I am trying to solve the following integral having two strictly positive parameters:
...
- 23
4
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2
answers
269
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Displaying integrals with non-variable factors in front [duplicate]
In class I sometimes go through the steps in mathematical derivations using Mathematica. Some of the steps involve substitutions and assumptions that come conceptually outside a strict mathematical ...
- 42.3k
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1
answer
49
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Resubstituting variables back into the calculated
I have lengthy second order derivative of a function that is defined with multiple lines of variables. When I compute the derivative how can I put back into the computed result the auxilary variables ...
- 1,630
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answer
90
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How to express the argument through the function?
I would like to express argument $d$ through $f$ and $B$ and to get $d(f,B)$
$$f(d,B)=B\left(\left(\frac{1}{4}-\frac{d}{B}\right)\left( 1-\left(1- e^{-30d/B} \right)^8\right)+\frac{2/B}{3+1/B^2}\left(...
- 1,893
2
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2
answers
302
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Series expansion using binomial theorem in Mathematica
The series expansion of $\displaystyle\left(1-\frac{a}{x}\right)^{1/3}$ using the Binomial theorem is given by
$$\displaystyle\left(1-\frac{a}{x}\right)^{1/3}=1-\frac{a}{3x}-\frac{a^{2}}{9x^{2}}-\...
- 179
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2
answers
63
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How to print particular cases with Integrate?
I would like to perform the following simplification:
...
- 254
1
vote
1
answer
125
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Evaluating a simple limit in Mathematica
Consider the function
$$M = \frac{\left[(f^{\prime})^2 - 2ff^{\prime \prime} - f^{\prime}_H~f^{\prime}\right]}{2f}$$
where $f \equiv f(r)$, and the primes are derivatives with respect to $r$. $f'_H$ ...
- 83
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2
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37
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How to simplify an expression with simple symbolic integrations with the same boundaries?
I am new to Mathematica and I am trying to simplify an expression with symbolic integrals as below
...
6
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2
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428
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Prove linearity of integration [duplicate]
I'd like to prove the linearity of integration over one real variable ($x$).
Integrate[f[x] + b g[x], x] == Integrate[f[x],x] + b Integrate[g[x],x]
which I was ...
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4
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702
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How to calculate the integral only in real domain?
I have the following integral:
Integrate[1/(Sqrt[2] + Cos[4 X] + Sin[4 X]), X,
Assumptions -> Element[X, Reals]] // Simplify
...
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56
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Replace original function inside its derivative
I'm trying to simplify a derivative calculation. I have the Zhs in function of x :
...
2
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1
answer
103
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How can I make sure that the two given numbers are exactly the same?
I have the given numbers $n1$ and $n2$. How can I make sure that these two numbers are exactly the same? Numerically, using {N[n1,40], N[n2,40]}, by increasing the ...
- 361
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2
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How to ask Mathematica to rewrite the larger arguments of sine function as a smaller number which is multiple of $\frac{2\pi a}{11}$
I have a trigonometric function $Exp$ where $\{f,g,h,k\}$ are some parametric functions, and $a\in\mathbb{N}$ and $x>0$.
$$ Exp=f \cdot \sin \left(\frac{58 \pi a}{11}+x\right)+g\cdot \sin \left(\...
- 1,159
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1
answer
73
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What is the most efficient/optimal way to simplify the given function? Does the given code need more assumptions?
I have a very long function for $\{x,b\}\in\mathbb{R}$ and $n=\{1,2,3,4,5,6,7,8,9,10\}$; here, I have only mentioned a short part of that. The function is a sum of the complex exponentials (a picture ...
- 1,159
2
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2
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160
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Simplifying a sum of complex exponential terms
I have this function for $n=\{1,2,3,4,5,6,7,8,9,10\}$. Is there any hope to significantly simplify this function (make it shortened) for general $n$? I use Simplify
...
- 1,159
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0
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89
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Simplifying the function for the common factors; and showing that it contains a multiplicative factor $i e^{i (6 x + 3 b + t)}$
I have the given long function for $\{x>0,b,t\}\in\mathbb{R}$ and $\{m,n\}\in\mathbb{N}$. I have two questions:
This function contains many common factors in polynomial form (as can be seen in the ...
- 749
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4
answers
114
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Simplifying a function to get rid of fractional powers of $-1$
I have the following function for $x>0$.
...
- 749
1
vote
1
answer
172
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Automated decomposition of a large symbolic matrix for faster determinant calculation
I have this $44$ dimensional parametric matrix ($\{x,d,b,t\}\in\mathbb{R}$) and I want to compute its determinant.
Questions:
Is it possible to predict how much time it will take to give a result by <...
- 1,159
4
votes
1
answer
98
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Bug in FullSimplify
Studying the generalized sum of the certain divergent series in version 13.1 on Windows 10 by
...
- 29.1k
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votes
0
answers
35
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Simplifying inner product of eigenfunctions, where eigenvalues satisfy a transcendental equation
I have obtained the following eigenfunctions as solutions to a PDE problem:
$$u_{n}({y})=\cos\mu_{n}\cosh\mu_{n}{y}-\cosh\mu_{n}\cos\mu_{n}{y}.$$
I also obtained the eigenfunctions to the adjoint ...
- 11
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0
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86
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How to simplify a formula with integrations by parts?
I have a huge expression to compute, which contains lots of different terms composed of multiplication of various functions, derived a certain number of times (non derived, derived once or twice). For ...
- 208