All Questions
Tagged with calculus-and-analysis assumptions
144 questions
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Problem with integrating special function under assumption
i found different results when integrating a special function (see below), depending where i place my assumption (x > 0, x0 > 0). The problem is that the two solutions are not compatible.
In fact, if ...
- 497
0
votes
0
answers
107
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Bug in Integrate with Assumptions?
I encountered behaviour I wasn't expecting with Integrate and thought I'd ask here before reporting a bug in case I'm just missing something obvious. A simple example is the following: Computing
<...
- 1
7
votes
2
answers
648
views
Forcing Mathematica's Integrate to give more general answers
I have a simple gaussian integral: $\int^{\infty}_{-\infty}dx\:e^{i\alpha x^2}$.
If $\alpha \in \mathbb{R}$, then:
$$
\int^\infty_{-\infty} dx\; e^{i \, \alpha x^2} = \sqrt{\frac \pi {-i \alpha}} \...
- 607
10
votes
1
answer
290
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Linearity Assumption [duplicate]
I'd like to define a set of variables of this sort:
{K1[t],K2[t],...}
Adding a general assumption: all such variables are linear.
I'd like to do this to avoid ...
- 2,202
1
vote
0
answers
62
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6
votes
1
answer
310
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Integrate under assumptions
I am trying to solve the following integral:
$$\int_0^\infty u^{\sigma-1} \exp\left[-c u^\sigma\right] \mathrm{d}u$$
under the assumptions $\sigma \in (0,1)$ and $c>0$.
I know the result is
$$\...
- 239
1
vote
1
answer
141
views
Can this integral be calculated using Mathematica?
I have the following integral:
$$
\int_{-\infty}^{\infty} \frac{\mathrm d\tau}{2\pi \mathrm i} \int_{-\infty}^{\infty} \frac{\mathrm d\tau'}{2\pi \mathrm i} \frac{\mathrm e^{-c^2(\tau - \tau')^2}}{(\...
- 217
1
vote
2
answers
127
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Integration using assumption in Mathematica
I want to reproduce the solution of the following integral, using Mathematica: $$\int_0^1duu(1-u)^2\frac{1}{(1-u)(1-v)}\left(\frac{1-u}{u}(1+\frac{1}{1-v})\theta(u-v)+\frac{1-v}{v}(1+\frac{1}{1-u})\...
- 13
1
vote
0
answers
46
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How to define a variable $x$ such that $Abs[x]<1$ and use this globally without having to re-write it again and again?
I am writing a code or a calculation of a very complicated function and my main variable is $x$ such that $-1<x<1$. This assumption has to simplify my final answer a lot, but since the final ...
- 215
3
votes
1
answer
311
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Is there a way to automatically find the limits of integration given a set of constraints?
I have a complicated function $F(x,y,z)$ which I want to integrate subject to these constraints:
$y\geq z\geq-y, y \geq 0, y \leq \frac{x+z}{3}$.
It's only a 3D problem, so it's possible to draw a ...
- 625
2
votes
0
answers
82
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How to express continuity assumption for Integrate?
I am wondering if it is possible, and how if so, to express function continuity in the assumptions for the definite form of Integrate.
First of all, while
...
- 794
4
votes
0
answers
131
views
Trigonometric integral with assumptions fails
Consider the following integral
Assuming[Element[{n, m}, Integers] && n >= 0, Integrate[Cos[ϕ]^n Exp[I ϕ m], {ϕ, 0, 2 π}]]
...
- 12.1k
2
votes
0
answers
96
views
Fourier Cos coefficient vs Integrate vs common sense
I was trying to find out what the (discrete) Fourier transformation of the following function is
$$
f(\nu, u) = \frac{\sinh \nu}{\cosh \nu - \cos u}
$$
So at first, I tried to evaluate a fancy MMA ...
- 2,396
0
votes
1
answer
104
views
Integration of a function with 3 parameters
Integrate[Y^(b - 1) (1 - Y)^(a - 1) (-Log [Y])^-k, {Y, 0, 1},
Assumptions -> {0 < a, 0 < b, k != 0}]
...
0
votes
2
answers
677
views
Defining all variables as real and positive before differentiation
I'm attempting to differentiate an equation in the form
D[sqrt((2*(((a*b*c+Pi*d*e^2+Pi*f*g^2+h*i*j+Pi*k*l^2)/(a*b*c+Pi*d*e^2+Pi*f*g^2))-1)*m)/(n^2 - o^2)/p), a]
...
- 21
3
votes
3
answers
592
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Incorrect definite integral
I have a definite integral that depends on a parameter. The integrand is nonsingular when $h$ is real and $> 1$. When I evaluate the integral with assumption $h > 1$, then Mathematica returns ...
- 31
2
votes
1
answer
208
views
Problem with SeriesData
Problem:
I need to find the leading order term in an expansion whose leading order behavior is a priori unknown. I can of course go with Series and try different orders, say ...
- 2,204
0
votes
0
answers
475
views
Integral of a step function ,
I am trying to integrate a step function given in a book, but it doesn't work.
...
- 123
0
votes
0
answers
50
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Is it possible to tell Mathematica that a user defined function is continuous? [duplicate]
If I use an arbitrary function f'[t] inside a definite integral, Mathematica (correctly) refuses to apply the fundamental theorem of calculus, because it doesn't ...
- 101
1
vote
0
answers
122
views
3
votes
1
answer
62
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Potential ambiguities in function Limit (Generate explicit conditions on parameters)
I evaluate the following expression with Mathematica 11:
L = Limit[-x^2/(y - Sqrt[y^2 + 2 x^2]), x -> 0]
This yields the incorrect answer $L=0$ rather than ...
- 65
0
votes
1
answer
138
views
Possible bug in integration involving $\cos (n \phi)$
I wanted to verify the integral $\int\limits_0^{2\pi}\frac{d\phi}{(a^2+b^2)+(a^2-b^2)\cos(n\phi)}=\frac{\pi}{ab}$ (where $b>a>0$ and $n$ is a positive integer).
Individual values of $n$ work:
<...
- 2,470
2
votes
1
answer
225
views
Integrate is not using Assumptions correctly
My question here is distinct to those that appear to have been asked earlier here, such as 19833.
The broad problem is that Mathematica is not simplifying the Assumptions given before usage. This is ...
- 195
6
votes
3
answers
1k
views
Integrating a piecewise function with symbolic pieces
Suppose I want to compute the antiderivative of this simple function:
$$f(t)=\begin{cases}
1 & \text{if $0\le t\le t_1$,} \\
-1 & \text{if $t_1\le t\le t_2$}, \\
0 & \text{otherwise,}
\end{...
user484
1
vote
2
answers
85
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Series -- one assumption works but a slight variation does not
Question
Consider the following two lines of code, which differ only by replacing 1 by y:
...
- 121
0
votes
2
answers
93
views
5
votes
2
answers
157
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How to integrate assuming some integrals are named constants?
Imagine having some undefined functions $X(x),\ Y(x), \dots$, and all we know about them is that they integrate to specific constants, i.e.
$$\bar X := \int_{-\infty}^\infty \text d x \ |X(x)|^2, \...
- 51
0
votes
1
answer
111
views
Finding a limit or solving with a limit?
I have a complicated equation that has a lot of terms with squareroots. I am simplifying that complicated function with assumption of that x << y , so that later I can use that solution to ...
- 893
2
votes
1
answer
389
views
Restricting the domain of the integration limit to be positive
I am trying to put a restriction on the limit of integration. I tried "Assumptions->" and Assuming[(){Refine[]}], but it doesn't work.. The integral I have is the following:
...
- 23
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votes
1
answer
2k
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Integrating a function from negative inf to positive inf with assumptions of variables
The function is $e^{a x^{2}}$ and to be integrated over the domain from negative infinity to positive infinity.
The variable a is to be positive using the ...
- 680
2
votes
1
answer
144
views
Strange results by integrating Abs[Sin[a - t]]
I have a (simple) integral with a unknown parameter a:
$$\int_0^{2\pi}|\sin(a-t)|dt$$
One can observe that it is just two times the area under sin curve from $0$ to ...
- 7,911
2
votes
1
answer
507
views
Calculating Limit with option Analytic->True and $Assumptions yields Series::cas
Overview
I am using Limit with the option Analytic->True to check the finiteness of some expressions with generic functions ...
- 725
1
vote
1
answer
175
views
How to find analytical solution for integral from product of Bessel functions of second kind with different order?
I need help with analytical solution for the following integral:
...
- 177
3
votes
3
answers
300
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How make D[f,x] treat OverBar[x] as a constant
I have an expression of the form f = a*x +OverBar[x], where OverBar[x] is just a name, it has not much to do with the variable x. I want ...
- 1,329
4
votes
1
answer
292
views
Integrate: Mathematica does nothing, not even repeat the code!
I need to calculate the following integral:
$$
\int_{-\infty}^{\infty}da\int_{-\infty}^{\infty}db\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}dy~e^{\frac{-(b^{2}+(a-10g)^{2})}{g^{2}-1}}e^{\frac{-(...
- 183
2
votes
1
answer
86
views
3
votes
1
answer
106
views
1
vote
3
answers
191
views
Incorrect sinc limit
I'm trying to compute in Mathematica 11.0 the limit of the $\mathrm{sinc}$ function at integers:
Assuming[k ∈ Integers, Simplify[Limit[Sin[π x]/(π x), x -> k]]]
...
- 233
2
votes
1
answer
644
views
How to properly input limit of real function of real variable? [closed]
Consider the following limit:
$$\lim_{x \to 0^+}(\ln x)^x.$$
When restricted to real functions (no complex version of $\ln x$ or power), then this limit does not exist because the function is not ...
user45207
12
votes
3
answers
639
views
Assumptions allowing to calculate an elliptic integral
When I feed Mathematica the following integral:
Integrate[Sqrt[(A - x) (B - x)/x], {x, 0, B}]
it spits it back out without evaluating it. However, it can ...
- 121
1
vote
1
answer
79
views
Assumptions not working with Trianglewave or Floor functions
The assumption below:
Simplify[Integrate[Sin[2 n π x], {x, 0, 1}], Assumptions -> n ∈ Integers]
simplifies the integration and gives $0$ as the correct answer....
- 529
2
votes
2
answers
363
views
How to treat indexed variables as Reals? Can they be symbolized?
First of all, this is quite related to this question here, however the solution given there does not apply to my issue.
I am generating a huge symbolic matrix, which has many (on the order of 50-100)...
- 2,732
0
votes
1
answer
88
views
Definite and indefinite integrations produce different results for same integrand
Consider a function subjected to the following assumptions:
...
- 177
0
votes
3
answers
822
views
Specifying multiple assumptions in the integration
I have to calculate following integral
$$ \int_{-\infty}^{\infty} \frac{d \omega}{(i \omega +(\mathbf{k}-\mathbf{p}).(\mathbf{k}-\mathbf{p}) ) (-i \omega + (\mathbf{k}+\mathbf{p}).(\mathbf{k}+\mathbf{...
- 109
-1
votes
1
answer
142
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Integration with Assumption, why doesn't it simplify? [duplicate]
I have a problem which surely someone else had on here, but in reading similar questions, I haven't found an answer.
The problem is really trivial:
...
- 963
2
votes
3
answers
325
views
Why do I get contradictory assumptions and how can I avoid them?
I'm trying to solve this integral
$$P\int_{-\sqrt 2 a}^{\sqrt2 a} dx \frac{\sqrt{1-(1-(x/a)^2)^2}}{x-y}$$
where P stands for Cauchy's Principal Value. So I code this
...
- 91
2
votes
1
answer
1k
views
Do double vertical bars in ConditionalExpression mean 'or'? [closed]
I have a very simple interpretation question. I think that double vertical bars in a ConditionalExpression given by Mathematica as the result of an integration must mean 'or', but I am hoping to ...
- 123
1
vote
1
answer
434
views
How to integrate with integer parameter in Mathematica?
I would like to integrate the following expression with N being Integer
...
- 155
0
votes
1
answer
2k
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Assumptions with Multiple Rules
What is the correct way to add more than one assumption while integrating?
I am attempting to evaluate the following integral and Mathematica is just stalling, so I'm wondering if my assumptions are ...
- 1,060
0
votes
0
answers
42
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