Questions tagged [calculus-and-analysis]
Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.
4,918
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Function asymptotics showing different result after inter integration [closed]
I have a function fg[r_, t_, t2_, \[Phi]_, DR_]. I want to do the following
Integrate it for t2 from 0 to t
Then take the asymptotic limit r-> 0
The result of ...
- 1
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1
answer
47
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Plotting an integral with solving for a specific constant [closed]
Given an Integral I(t) which is a function of a constant tc, this constant is determined with the condition I(tc)=0, how can I add this condition?
- 117
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1
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47
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Numerical Integration without giving all values
Is there a way to solve expressions numerically, while keeping other variables.
For example, I want to Integrate the expression below for $\theta p,\theta q$ while keeping other things as it is. The ...
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2
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1
answer
43
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How to linearize these terms in many different variables?
How to linearize these terms in q[t] and p[t,y]
...
- 209
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41
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Seeking Wolfram Mathematica Techniques for Detailed Step-by-Step Solution of Integral Equation Transformation
Hello Mathematica Community,
I am working on a mathematical problem where I need to transform an integral equation into a simpler form using Euler's Formula and the Error Function. My current ...
- 1
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0
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42
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Derivative of a function in euclidean distance [closed]
differentiate W in (Z=XW, D1=distance(Z[, 1]), D2=distance(Z[, 2]), D3=distance(Z[, 3]) exp[-D1/theta _1 -D2/theta _2 -D3 /theta _3]) where W is 5 by 3 matrix, Z is 100 by 3 matrix and X is 100 by ...
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0
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44
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How do I find roots of a third degree polynomial? [closed]
expr = G^2 X^3 -
2 Δ G X^2 + Δ^2 X + ((ℏ *G*η*
k*Subscript[E, l]^2)/(m*Subscript[Ω, m]^2)) == 0
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2
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75
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Integrate with `Max` and `Min` under assumptions runs forever
I'm trying to solve an integral under certain conditions on the parameters. The following is my code.
...
- 979
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1
answer
52
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Leading significant digits
Given that the initial area in a Sierpinski triangle is A_0 and the subsequent areas
$$(A_n) =A_0 (3/4)^n$$ ,
what is the formula to calculate the leading ...
0
votes
1
answer
73
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What does "{True,True}" mean in the result of IntegrateChangeVariables?
I want to change variables in the following integral:
Integrate[q*D[z[x, y],x]^2, {x, -∞, ∞}, {y, -∞, ∞}]
as follows:
...
- 38.7k
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98
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Integral of complex exponential with assumptions
I feel confident this problem has been posed before but alas I cannot find it.
...
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39
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2
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3
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141
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Find Generalized Series with Symbolic Variable
CoefficientList[Series[Exp[x], {x, a, 3}], x]
Gives the following expression,
$$
\left\{-\frac{1}{6} e^a a^3+\frac{e^a a^2}{2}-e^a a+e^a,\frac{e^a a^2}{2}-e^a a+e^...
- 123
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49
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Express the double integral in terms of the function values at discrete nodes
Consider the following integral:
$$
\mathcal{I}[p_{1},t] = \int \limits_{0}^{p_{1}}dp_{2}\int \limits_{0}^{p_{2}}dp_{3} \ \text{integrand}
$$
where
...
- 4,642
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71
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Lagrangian density and Christoffel symbols [closed]
According to this article (equation 3.2) the Lagrangian density can be calculated by
$$ 2 \mathcal{L} = \left\{\begin{array}{c}
a \\
b c
\end{array}\right\}(g b c \sqrt{-g})_{, a}-\left\{\begin{array}{...
- 103
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votes
1
answer
58
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How to linearize this equation in two different variables?
How to linearize this equation in two different variables p[t] and q[t,y]:
...
- 209
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vote
1
answer
53
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First order differential equation with two conditions
I want to produce all possible solutions for this equation:
...
- 153
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46
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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 ...
- 661
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0
answers
106
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Integrate function differing from RUBI's Int function in version 13.3.0
I had Mathematica Version 12.3.1 Student Edition and Integrate and RUBI's Int functions produced the same answers for my purposes with good results. I updated my version to 13.3.0 Student Edition and ...
- 9
1
vote
1
answer
100
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Plot after applying NIntegrate
I'm trying to integrate and graph $I$ vs $b$ the following integral (eq. 24 in this paper) in wolfram mathematica
$$ I = \int_{2}^{\infty} \frac{g_i^3 \mathcal{K}_t \mathrm{~d} r}{r^2\left|\mathcal{K}...
- 103
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1
answer
84
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0
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1
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91
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Change of variable of a numerically solved integral
I have the following function that numerically integrates a function on the interval $\left[ 2, 100 \right]$
...
- 103
1
vote
0
answers
81
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Intensity pattern [closed]
I'm trying to get the following intensity pattern from this paper
In the Newtonian case, the equations for the intensity (eq. 9 and 10 in the article), I put them in Mathematica with the change $b^2 =...
- 103
1
vote
1
answer
87
views
Definite integral of real function outputting complex values
I'm trying to integrate the function $f(\omega)$ from $-\infty$ to $\infty$, where
$$
f(\omega) = \begin{cases}
\frac{(1-2 \omega )^2}{64 \cos (\tau (-\omega +\phi
+1))-80} & 0<\omega \leq ...
- 111
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votes
1
answer
47
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How to chose appropriate finite limits for NIntegrate?
I want to NIntegrate a function using finite limits so it gives a result as close as possible to a result which I would get making the limit ...
- 55
1
vote
1
answer
257
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Plot solution of an integral with parameter
In this article, they solve the following integral
$$ \mathcal{I}_{o b s}\left(\nu_0\right)=\int_{\gamma_i} \frac{(A(r))^{3 / 2}}{r^2} \sqrt{\frac{1}{A(r)}+r^2\left(\frac{d \varphi}{d r}\right)^2} d r ...
- 103
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votes
2
answers
85
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How to prevent evaluation without simplification?
I am trying to compute a sequence of terms, which have a form similar to $g(z)$ at different points $z$. For $z$ close to 0, I have observed that the terms are evaluated as 0/0 (indeterminate form) ...
- 121
8
votes
3
answers
3k
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Why doesn't Mathematica recognise two integrals as equal?
The following expression
$$\int_{0}^{t} f(t_0) \,dt_0 - \int_{0}^{t} f(t_1) \,dt_1 = 0$$
is zero because it is just a change of integration variable.
Why doesn't Mathematica give zero in this case?
<...
- 83
4
votes
1
answer
182
views
Implementation of SINC filter using Integrate results in incorrect output
Bug introduced in 6.0, persisting through 13.3.1.
This is SINC filter, integrate $|H|^2$ from $-\pi/2$ to $\pi/2$.
MMA CODE, Result = 7.27956e-12
...
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1
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2
answers
52
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How to find the roots of an equation for different wave numbers?
I have a equation nn which is a function of t and k. I want to find the maximum of the ...
- 349
1
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2
answers
186
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Sum $\sum _{k=0}^{\infty } \frac{(-z)^k}{\text{IteratedLog}(k+2)}$ [closed]
The following result can be obtained by means of succession acceleration. Can this result also be calculated in Mathematica?
$$\sum _{k=0}^{\infty } \frac{(-z)^k}{\text{IteratedLog}(k+2)}=\frac{z+2}{2 ...
- 43
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1
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132
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0
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0
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59
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Plotting list of points to a special graph
Given Mathematica code of eigenvalues in different variable-paths , I want to add experimental data in the path Gamma-X ; X-Gamma ; and M-Gamma .
Here is the code :
...
- 117
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0
answers
271
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Symbolic integral with UnitStep functions
I want to symbolically evaluate the following integral (with real positive parameters L1, L2, ...
-2
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0
answers
68
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How to make NIntegrate with large integration limit?
I try to double integrate this equation with respect to $b$ and $f$
...
- 209
1
vote
1
answer
78
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Minimization of an implicitly specified function
exp - an experimental data, where expT - temperature, expChi - magnetic susceptibility
I ...
- 1,753
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votes
1
answer
58
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Syntax error trying to rewrite expression using rules
I want to use Mathematica to calculate derivatives of some long, complicated expressions. I have an integral with integrand $I_*$ which is a functional that depends on $h(x,t)$ and $\phi(x,t)$. The ...
- 221
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votes
1
answer
70
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Divergent series $\sum _{k=0}^{\infty } (2 k+1)^{3 k+1} z^k$ and LambertW function [closed]
I can get values from the following function,
$$\sum _{k=0}^{\infty } (2 k+1)^{3 k+1} z^k $$
could you tell me how you could check those values using the Lambert function in integral form or if you ...
- 43
1
vote
1
answer
151
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Why does Mathematica fail to compute $\int_{0}^{\infty} \sin(ax^2) dx$ whereas Wolfram Alpha seems to be able to do it? [closed]
I was symbolically computing
$$ \int_{0}^{\infty} \sin(ax^2) dx $$
Via the command:
Integrate[Sin[a*x^2], {x, 0, Infinity}]
But it seems to fail with mathematica, ...
- 133
0
votes
1
answer
50
views
Different integral values for analytically same integrals
Consider a function $L$ with parameters $h=0.1,\ D=1.5$
$L(m,x):=e^{-\frac{2\ (\frac{x}{h}-m)^{2}}{D}}$.
While the integrals $\int_{-\infty}^{+\infty}L(0,x)dx$ and $\int_{-\infty}^{+\infty}L(40,x)dx$ ...
- 3
0
votes
3
answers
137
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Finding the area enclosed by two curves
I am trying to find the area enclosed between two curves (i.e., two datasets).
The challenge is to estimate the cross-sectional area.
as indicated in blue.
The first dataset represents the bathymetry ...
- 587
3
votes
6
answers
447
views
Galactic rotation speed
Rotation speed vc[R] in reduced units for M33 looks like
0.342331 Sqrt[R^2/(0.728117 + R^2)^0.87955]
Knowing ...
- 193
2
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0
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142
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Finite sum of complex exponentials giving wrong result [closed]
I was playing with the following sum:
$$ \sum_{k=0}^{2} e^{\frac{2\pi i k^2}{3}} = e^{0} + e^{\frac{2\pi i}{3}} + e^{\frac{8\pi i}{3}} \ne 0, $$
which is non-zero, at least according to Wolfram|Alpha.
...
- 133
2
votes
2
answers
88
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How can I find maximums of a function without Table and FindPeaks?
I have the following equation
$$\pi_{xx} = \exp(-2 k_x^2 \sin^2(\frac{t}{2})) k_x^2 \theta(t)
\times \bigg[ -\frac{\sin(2t)}{2} - \cos(t) \sin(t) (1 - k_x^2 \sin^2(\frac{t}{2}))
+ 2 ...
- 349
1
vote
2
answers
84
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How can Mathematica merge $(1/t)^{1/3} t^{17/3}$ in an expression?
(1/t)^(1/3) t^(17/3)
How to let Mathematica simplify it?
- 11
1
vote
3
answers
206
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Integration giving unknown messages
I am trying to integrate the following
Integrate[(Log[1 - t] t^(-(1/2) + eps))/(1 + t),{t,0,1}]
To simplify this I am doing a further transformation ...
- 1,604
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votes
2
answers
79
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How to find the maximum of an equation based on a variable? [duplicate]
I have the following periodic function, as a function of K and t. I need to know what is the maximum of each recurrence as accurate as possible for k={10,20,40,60,80}
The equation is as follows,
$pixx ...
- 349
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0
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106
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Why does Area produce finite values for this area? [closed]
Let us consider in 13.3.1 on Windows 10 the region
reg = ImplicitRegion[{x*y >= 8, y >= 8 x^3, y <= 27}, {x, y}];
and its area
...
- 24k
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0
answers
21
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The relationship between dependent decision variables in a function
Given objective function f(x1,x2,x3) in a optimization problem, and don't know if all these variables are independent or not. How to know if dx2/dx3 is greater/less than/equal to zero. I could
1
vote
2
answers
252
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How to interpolate an implicit function?
chi(t) is implicit function. The implicit function can be plotted by ContourPlot.
Is it possible to obtain chi(t) explicitly using function interpolation?
...
- 1,753