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.
2
votes
1answer
48 views
Singular Integration
I am trying to simplify the following integral but getting no answer. Any help in how to get the resulting function as a function of t would be much appreciated?
...
1
vote
0answers
47 views
Getting Mathematica to supply derivatives in standard format [duplicate]
I will like to use the results of Mathematica outputs in formatted text for students. However, I do not know how to get rid of the variable arguments and the use of raised indices to indicate ...
1
vote
1answer
70 views
An optimization problem over two variables
I have the function fun[x_,y_, t_] = Cos[x] + Exp[I y] Sin[x] Sin[t]; such that $0\le {\rm x}\le \pi$, $0\le {\rm y}\le 2\pi$, and $0\le {\rm t} \le 10$. I compute ...
0
votes
5answers
47 views
Expressing D[1/D1, x] in terms of D1 when D1 = 1 + x
For simplicity, taking D1 to be equal to 1 + x, I want to evaluate the derivative of 1/D1 (...
0
votes
1answer
41 views
Integrating in polar differs from simple integration [on hold]
I have same line in orthogonal and polar coordinates, it is Cos[t] in first quadrant. But when I integrate those, result differs and I dont understand why:
...
2
votes
0answers
48 views
0
votes
0answers
58 views
How to Solve This Integral?
Mathematica insists that the below integral does not converge, though it has been solved by my peers, so I am inclined to think that it does.
q is an arbitrary variable.
...
0
votes
1answer
22 views
Limit function and DirectedInfinity as output: Interpretation
I tried to compute the following limit:
In[12]:= Limit[-A/2 x - 2 x, x -> +Infinity]
Out[12]= DirectedInfinity[-2 - A/2]
Does the output tell me that the ...
-4
votes
0answers
69 views
Mathematica evaluates D[x, Log[x]] as 0 without raising any error [closed]
I thought the answer should be
$ \frac{\partial x}{\partial \ln{x}} = \frac{1}{\frac{\partial \ln{x}}{\partial x}} = x $
Why Mathematica 11.3 does not know that it made a mistake?
One possible ...
0
votes
1answer
78 views
DiracDelta is seldom chosen as the answer [closed]
It is well known that
Limit[ ϵ/π/(x^2 + ϵ^2), ϵ -> 0]
is the Dirac Delta function
DiracDelta[x]
Yet Mathematica gives ...
1
vote
1answer
33 views
Using rule with derivative
Assuming I have an expression like:
D[a[t, r], r] + D[D[a[t, r], r],r] + D[D[a[t, r], t],r]
I would like to change variables, by expression everything in terms of ...
1
vote
1answer
59 views
Very different results of NIntegrate using different methods: which one to believe
Consider the following function:
...
7
votes
2answers
741 views
Integral with DiracDelta. Can Mathematica be made to solve this?
This was an exam question.
$$
\int_0^{2 \pi} \delta(\sin^2(\theta) -x ) \,d\theta
$$
Direct use of Integrate on it does not give the solution. Is there a ...
2
votes
3answers
164 views
Mathematica won't evaluate this integral
The function (in raw input form) and bounds I used are as follows:
...
0
votes
2answers
97 views
Difficult limit [on hold]
I try to calculate the following limit to calculate the constants C[1] and C[2]:
...
0
votes
0answers
42 views
Different integration results when TrigToExp used
I am trying to do a double integral over a domain $x \in (\theta_1,\theta_2), y \in (0,\theta_1)$. Instead of doing a definite integral I am doing an indefinite integral and taking limits. All that is ...
0
votes
1answer
46 views
How do I compute the infinity-norm of a function? [closed]
I have a technical question on Mathematica 11.
I have to compute the $\infty$ norm of q function, $f(x,y)$, on a rectangular domain $[a, b] \times [c,d]$. How can I do this with Mathematica?
3
votes
2answers
442 views
1
vote
1answer
42 views
How does one test whether or not an integration has succeeded?
This must be a very simple matter--but how to address it eludes me at the moment.
I have a series of integrands--and want to flag whether the symbolic integation has succeeded or not (in order to ...
0
votes
3answers
108 views
Limit of this definite integral [closed]
How can I compute this integral for $y\rightarrow0$ ?
$$ \int_0^1{\frac{y(1-x)^2(1+x)}{x+(1-x^2)y} dx} $$
0
votes
1answer
96 views
Numerical integration giving trouble
I am trying to do the following integral numerically,
$$\rho(\theta_{j},\phi_{j})=\int\frac{\sin{\theta_{i}} d\theta_{i} d\phi_{i}}{\sqrt{2+2[\cos(\theta_{i})\cos(\theta_{j})+\sin{\theta_{i}\sin{\...
2
votes
2answers
623 views
Weird result in complex limit
I am trying to evaluate a limit:
gamma[w_] = Sqrt[-(u*e)w^2 + I*(u*s)w];
Limit[Re[gamma[x]], {x -> DirectedInfinity[1]}]
I calculated the limit by hand, and ...
1
vote
1answer
28 views
Verify that a PDE is satisfied by a matrix consisting of functions satisfying a different PDE
My problem is as follows: I have two matrices $P$ and $Q$ which are made up of algebraic expressions of a function $q(x,t)$ which solves a somewhat complicated pde, and its derivatives, for example, ...
0
votes
2answers
76 views
Integration gives a complex valued result
There are many similar questions but they all seem to be integrating functions that are more involved than just a product of polynomials and the issue has more to do with the integrand than the ...
1
vote
1answer
55 views
How can I solve this integration or does it have a closed form solution?
The integral I am dealing with is below.
I need to find the closed-form expression of this integral.
$\int_0^\infty \ln(1+\frac{A}{1+B+Cx})\frac{e^{-x/M}}{M}dx$
Here, $A$, $B$, $C$ and $M$ are ...
1
vote
1answer
67 views
Centroid of a cosine shaped rod
There's a problem in Hibbeler's 14th edition of Statics for Engineering that I've tried a few times and couldn't solve. The problem is shown below. It wants me to find the centroid coordinates of a ...
10
votes
2answers
803 views
Why does Arg'[1. + I] return -0.5?
From the document we know that
Arg[z] gives the gives the argument of the complex number z.
Then how about ...
3
votes
3answers
189 views
Approximating integral with small parameter
I want to approximately compute integral $$I =\int_0^1 dx \frac{x(2-x)(1-x)}{(1-x)^2+\mu x}$$ assuming that $\mu$ is small. I tried
...
0
votes
1answer
45 views
integration of discontinuous function [closed]
I am sure this is an easy question. Although I am new in mathematica. I want to integrate a function that has some (removable) singularities.
$ f_h(x) = -x\cot{h\pi x} - \frac{1}{\pi h}\sum_{j=1}^{h} ...
0
votes
0answers
35 views
Calculate a definite integral containing an ExpIntegral
I need to calculate the following integral
Integrate[Exp[-(y/bn)] (-y)^c ExpIntegralE[d, p y], {y, 0, z/x}]
Where $z<0,z\in \mathbb{R}$, $x\in\mathbb{R},x>...
1
vote
1answer
45 views
Closed form solutions to functions involving Erfc
Is there a better way to get a closed form solution in terms of sigma? I've already attempted a significant amount of simplification up to this point and am unsure ...
0
votes
1answer
65 views
Solve a constrained four-dimensional integration problem for a “magic simplex” probability
Equation (7) in the 2012 paper, "Complementarity Reveals Bound Entanglement of Two Twisted Photons" of B. C. Hiesmayr and W. Löffler for a state $\rho_d$ in the "magic simplex" of Bell states
\begin{...
3
votes
2answers
740 views
How to evaluate this function?
I’m trying to evaluate
y[t] = InverseFunction[ NIntegrate[Sqrt[2]/( -8+Exp[y]+2Exp[-y]), {y,0,t}]]
Some of the problems I’m facing, is that i’d like to get the ...
0
votes
1answer
80 views
Substitution into integral [duplicate]
I have to do a substitution (u = pi/2-x) into an definite integral, which I have defined as f[u]. When I evaluate my code, no ...
1
vote
1answer
49 views
Replace second derivative with product rule
Suppose that after some calculations, I find
a[x]*D[g[x],x,x]+b[x]*D[h[x],x,x]+c[x]*D[j[x],x]
What sort of operations do I need to do in Mathematica such that ...
0
votes
0answers
92 views
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
<...
3
votes
3answers
95 views
Evaluating the error function by integrating $ \mathrm e^{−t^2} $ with Simpson's rule
I am trying to evaluate an error function with Simpson's rule because there is no other way to integrate it. The function is
$$
{\rm erf}(x) = \frac{2}{\sqrt π}\int_0^x \mathrm e^{−t^2}\, \mathrm dt
...
2
votes
3answers
118 views
Integration Help
So I have to integrate $$\frac{\sin^n x}{\sin^n x + \cos^n x}$$ and am coding this in Mathematica with
(((Sin^n)[x])/(((Sin^n)[x]) + ((Cos^n)[x])))
with the ...
2
votes
2answers
136 views
Poor/Incorrect handling of derivatives of DiracDelta functions in Mathematica
Bug introduced in 12.0 or earlier
Integration of $\int_{-\infty}^{\infty} e^{-2|x|}\delta''(x) dx = 2 \int_{-\infty}^{\infty} e^{-2|x|}\frac{\delta(x)}{x^2} dx \rightarrow$ Divergent!
Now in ...
0
votes
0answers
43 views
Mathematica does not compute derivative of composite function
I was checking some stuff from my Mechanics Course (non-inertial systems) using Mathematica and I realized I can't get it to compute the second derivative with respect to time.
If you could tell me ...
-1
votes
1answer
87 views
Plot fractional trigonometric functions with the Mittag-Leffler function
Can anyone help please?
Im trying to plot the solution $X$ of the system as in the paper attached - about fractional calculus which is
$X= [E_{\nu}(2t^{\nu})][2 \cos_{\nu}(3t^{\nu})+4 \sin_{\nu}(3t^{...
3
votes
2answers
106 views
Relations between two reciprocal partial derivatives?
My question is similar to How to get the partial derivative of the inverse functions?
But they are different.
If we have a function $z=z(x,y)$, we can calculate the partial derivative $\left.\frac{\...
3
votes
3answers
319 views
Finding the area between two curves with Integrate
I'm trying to solve and approximate the area between the two graphs. Right now, my functions are stored as
f[x_] := 3 Sin[x]
g[x_] := x - 1
and then I tried to ...
0
votes
1answer
78 views
Plotting with the Mittag-Leffler function [closed]
I'm trying to plot the solution of fractional differential equations as shown in the photos below, The solutions are in terms of the Mittage-Leffler function, so I evaluated
...
3
votes
0answers
108 views
Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$
Saw this post and noted that it was posted back in 2013. Just tried V11.3 and got no analytic solution.
Just wondering if there is a way for Mathematica to give the desired result?
Machine VS Human
...
1
vote
1answer
81 views
Definite Integration by parts
how do I code
Integrate[u[x] D[v[x], x], {x, a, b}]
to return the following straightforward undergraduate result?
...
1
vote
1answer
73 views
Converting .STL from Shell to Solid for stress analysis of wing
I'm new to Mathematica as this is the first class I've had to used it in my undergrad time as an aerospace engineer. Anyway, I have run into a problem that I can't figure out. I have created .STL's of ...
1
vote
3answers
76 views
How to take a derivative of a function with a real part taken
Consider
f[x_]=Re[x^3]
I now take the derivative of this
D[f[x],x]
and the result is
...
1
vote
2answers
45 views
Series expansion of integral
I'm looking for a way to do a series expansion of
$$\frac{\mu_0bI_0}{4\pi}\int_0^{2\pi}\frac{\cos\left[\omega\left(t-\frac{1}{c}\sqrt{r^2+b^2-2rb\sin(\theta)\cos(\phi)}\right)\right]}{\sqrt{r^2+b^2-...
2
votes
1answer
54 views
How to make a function depend on the solution of an integration?
I have two functions which are the linear and non-linear solution of an equation. I'm trying to calculate where does the linear solution is a good solution of my equation, when compared with the non-...
