# 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.

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### 2-loop diagrams dimensional regularization with Feyncalc

I attempted to evaluate the following diagram using FeynCalc, following the method outlined by Pierre Ramond in his QFT book, but I encountered difficulties: \Sigma(p)=\frac{\lambda^2\...
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### Uncaught SystemException returned to top level calling integrate in V 14.1

Report to WRI [CASE:5169570] May be someone with better debugging skills than me in Mathematica internals could figure what causes this. This integral worked OK in V 14.0 but now in V 14.1 it gives <...
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### Simplifying an ODE in Bessel notation

I have the following Mathematica code: eq[r_] = 3 (1 + 2 a) g1 r y[r] - 2 (2 y'[r] + r y''[r]); DSolveValue[eq[r] == 0, y[r], r] This gives me the ...
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### Closed form of an integral using Mathematica or otherwise

Define $$I=-\int_0^1\int_0^1 \frac{x^2(1-x)y^2(1-y)(2(1-xy)+(1+xy)\log(xy))^3}{(1-xy)^7}\ dxdy$$ Now using Wolfram Alpha $I\approx 0.00133186$. Using Mathematica or otherwise, I need to find a ...
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### Best way to implement the "plus-distribution" for very long expressions?

I'm at my wits end here. Is there an efficient way to implement the following distribution into MMA: $$\int_0^y dx \frac{f(x)-f(y)}{y-x}$$ where $f/1-x$ can be analytically integrated. The difficulty ...
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### How to fix this "wrong" analytical integral evaluation?

Integrate[(1/Sqrt[(x + 1)^2 + 1])*(1/Sqrt[1 + x^2]), {x, 0, 1}] yields ...
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### How to use mathematica for differential geometry

I would like to learn how to efficiently use mathematica for differential geometry and tensor analysis. I googled it and mostly found some old references to packages. I wonder if there is a good ...
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### Integration or numerical integration over one variable out of many variables

I have a function defined as AFB = (6*I6parSMM)/(2*(3*I1cparSMM + 6*I1sparSMM - I2cparSMM - 2*I2sparSMM)); where the I'parSMM's are functions of other variables s, ...
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### I want to plot the group delay of the following mathematica code but it takes too long time

I am working on a Mathematica project where I want to plot the group delay. The problem is this code is running too slow. Is there any possibility of speeding up the code? Here is the original code: <...
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### How to calculate the average stream velocity from the segmentation of video frames?

I have a 10 second video of a river flowing. I want to calculate the velocity of the river flow using optical flow. In this operation, I used the ImageDisplacements ...
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### How can I find a condition for convergence of an integral? [closed]

Wolfram|Alpha says that $$\int_0^\infty \frac{x^n}{(x-1)(x-2)} dx=\pi (1-2^n)(\cot(\pi n)+i)$$ for $\text{Re}(n)>-1$. However, for $n=0$, it says the integral doesn't converge, even though it has ...
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### Represent and evaluate an infinite expression

I am trying to represent this in Mathematica, then evaluate it: ...
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### Integrate result differs from what's expected

Trying to get a result for this Indefinite integral: Integrate[1/(2*(x^4 + x^2 + 1)), x] Mathematica v14 provides this result: ...
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### Mathematica is giving me an error when I try to see what the form of a DifferentialRoot [closed]

I'm trying to solve a coupled system of PDE's. After some Fourier transforming, I was able to eliminate derivatives with respect to one of my variables which allowed me to eliminate one of the ...
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### Difference between the two forms of derivates

Can anyone explain the difference between these two forms of derivatives: f^(1,0) and D[f,x] where ...
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### Reliability of DSolve to ensure existence of analytic solutions

So I've been wondering about this for a long time. Of course, Mathematica DSolve can be a monster slayer sometimes but as Mathematica as it can get, the output results usually tend to be not ...
I need to perform this integration as a contour integral over the lower half of the complex omega plane with Mathematica. $$\int_{-\infty }^{\infty } \frac{e^{-i \left(\omega -\omega _0\right) (t-\tau ... 1 vote 2 answers 70 views ### Speed up definite integral of 1/x^(n/2) I'm a bit buffled by how terribly slow this is: ... • 1,441 1 vote 2 answers 67 views ### Why is the result of this integration coming out as a list if the function being integrated just returns a numerical value? I am trying to integrate over a Dirichlet distribution. Instead of using the built-in function, I defined it myself as: ... 3 votes 1 answer 69 views ### Many contradictory results for a single integral I am interested in solving the integral$$ \int_{\mathbb{R}} dx \frac{e^{i a x^2}}{(x - b - ic)^d(x - b + ic)^d} $$for a,b,c \in \mathbb{R} and d \in \mathbb{N}. As an early step, I have asked ... • 131 1 vote 1 answer 68 views ### Is there a more efficient way to calculate the variance of a random variable given two sets of data? Given two lists, one being the values of a random variable and the other being their corresponding probabilities, how can we calculate the variance of the random variable? ... • 4,707 1 vote 1 answer 82 views ### Incorrect limit involving hyperbolic Kepler equations I’m trying to work with the Kepler equations for a hyperbolic trajectory in the case of a repulsive force (see Wikipedia for the attractive case). The trajectory can be described by the right branch ... • 11 0 votes 0 answers 76 views ### How to Handle First Variation with Symbolic Computation in Mathematica? Background I am currently working on a problem related to surface tension and found a research paper that appears to have the information I need. I am attempting to replicate the calculations in this ... • 103 2 votes 2 answers 100 views ### Why does Replace function not work the first time I try it? Please explain why in the following mathematica code, the cos term was not replaced in the first try, but got replaced in the second try? ... • 121 2 votes 1 answer 93 views ### How to handle undefined Integral results for excluded conditions in Mathematica I have a mathematical function where both input parameters x and y are real numbers within the interval [−π,π]: ... • 1,648 0 votes 1 answer 45 views ### LineIntegrate seemingly does not work The scalar line integral function seemingly does not work, the both expressions remain unevaluated: ... • 3,662 3 votes 1 answer 122 views ### How to run Integrate[Sqrt[(x - a)^2 + (y - b)^2 + (z - c)^2], Element[{x, y, z}, Ball[{0, 0, 0}, 1]], Element[{a, b, c}, Ball[{0, 0, 0}, 1]]] I find that MMA cannot run the integral below ... • 941 0 votes 0 answers 41 views ### Issue solving recurrence relation I´m trying to solve the following recurrence relation: ... • 145 1 vote 1 answer 41 views ### Allan Variance integral for a first order Markov process I am using Mathematica to solve this definite integral: 4*(q*T)^2*Integrate[(Sin[Pi*f*t])^4/(Pi*f*t)^2/(1 + (2*Pi*f*T)^2), {f, 0, Infinity}] T and t are real ... 3 votes 2 answers 302 views ### 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: ... • 145 0 votes 0 answers 78 views ### Troublesome output of summation's derivative I am trying to compute a derivative of a nested sum, but I can't avoid Mathematica to output Kronecker deltas and unexpected cases on complex numbers. Consider the following setting: ... • 3 2 votes 2 answers 143 views ### Convergence of \left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\} as n\to \infty using Mathematica or otherwise Using Mathematica or otherwise, I need to find the convergence or divergence of$$\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$$where \{x\} denotes the fractional part ... • 219 1 vote 2 answers 117 views ### NDsolve does not solve equation I have a question for system of ordinary differential equations, because Mathematica does not give the solution. To be more specific, it does not give any sign that there is some problem. There is the ... 1 vote 1 answer 120 views ### Integration of Gaussian functions takes ages I am trying essentially to compute a fourier transform with Integrate. 1]Is integrate faster or slower than FourierTransform? 2]Does anyone know how to efficiently calculate these integrals? My code ... • 11 1 vote 1 answer 172 views ### What methods can quickly determine the coefficient of determination of the data? The given data is: x = {0.04, 0.06, 0.04, 0.08, 0.08, 0.05, 0.05, 0.07, 0.07, 0.06}; y = {0.25, 0.4, 0.22, 0.54, 0.51, 0.34, 0.36, 0.46, 0.42, 0.4}; Is there a ... • 4,707 3 votes 0 answers 131 views ### Trivial Mathematica integral on a cube not computing I was trying to evaluate some intergal over some cube with variable side length, but for some reason Mathematica spit out the input. The interand was sufficiently nice that I couldn't believe it didn'... • 208 5 votes 3 answers 735 views ### Mathematica integral wrong I need to evaluate the following integral: ... • 208 0 votes 1 answer 60 views ### Finding an increasing or decreasing function I have a problem where I am given the derivative of a function, asked to graph it, then state where it is increasing or decreasing using Mathematica. f'(x)=(x^2-3^x)/2x 0<x<4 I've tried the line ... 1 vote 2 answers 153 views ### Confused about the output of CosIntegral I am interested in the properties of the cosine integral function, in this case the anti-derivative of Cos[Pi*x]/x, which Mathematica evaluates to ... • 2,335 0 votes 1 answer 49 views ### Checking the sign value of the derivatives of a complicated function Consider the following function: ... • 765 0 votes 2 answers 80 views ### Integration with respect to functions I observe some strange behaviour of Mathematica when pulling functions into the integrator. I know that Mathematica's capabilities are limited when computing Stieltje or Lebesgue Integrals. But even ... • 187 4 votes 3 answers 258 views ### How is the result of this integral obtained by the function Integrate? When I use Mathematica to calculate$$ \int_0^{+\infty } \frac{\log (1-\mathrm i x)}{1+x^2} \, \mathrm dx , $$I get different results using functions NIntegrate ... • 411 2 votes 2 answers 96 views ### Partial Derivative after Numerical Integration of a Complicated Expression with Singularity at Zero I want to generate Fig no. 2,3,4 from https://arxiv.org/pdf/2212.08237 with the help of the following equations. In Fig 3 t_(opt) is the time Q_T reaches its max value Q_T(opt) for each T. Here is ... 1 vote 1 answer 68 views ### StreamPlot3d for the magnetic field of a loop I am trying to go beyond StreamPlot in the case of a current loop that produces a magnetic field. My code works for StreamPlot, but does not end when I use the same field expressions with StreamPlot3D.... • 43 3 votes 3 answers 356 views ### Integration of the product of two exponential functions How can I find the value of this integral? Integrate[E^((-b)*t^a - k*t), {t, 0, Infinity}] • 982 0 votes 1 answer 46 views ### Integrate diverges for convergent integral, and returns unnecessary condition I'm trying to solve an infinite integral using Mathematica. For some reason, Mathematica claims the integral diverges although when solving to a finite limit L and ... • 433 2 votes 1 answer 37 views ### Describing the region of integration \mathbb{R}_+^n for arbitrary n Let f_n:\mathbb{R}^n \to \mathbb{R}. I would like to know an elegant way of writing the integral$$ I_n = \int_{\mathbb{R}^n_+} f_n(x)~\mathrm{d}x $$in Mathematica. For fixed n, the best I could ... • 123 0 votes 5 answers 371 views ### Possible bug in partial derivative [closed] In 14.0 on Windows 10 let us consider D[Min[x, y], x] Piecewise[{{1, x - y <= 0}}, 0] As I understand it, the result is ... • 27.4k 9 votes 8 answers 754 views ### Numerical approximation of the integral by using data I want to use the numerical approximation of the integral of a function given a list of data:$$\int_a^bf(x)dx\approx\sum_{k=1}^N\frac{f(x_{k})+f(x_{k-1})}{2}(x_{k}-x_{k-1}), where $f(x_0)=f(a)$ and ...
I want to calculate an integration, which reads where $\delta\left(q_{23}^{01}\right)=\delta\left(1+q_1-q_2-q_3\right)$ and $\mathrm{min}(1,q_1,q_2,q_3)$ means the minimum of $(1,q_1,q_2,q_3)$. What ...