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Questions tagged [symbolic-computation]

Numeric computation usually uses floating point numbers. Symbolic computations use symbols, and can give exact answers, such as $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$. Mathematica, Maple, and Geometry Expressions all use symbolic computation, when desired. An online source is WolframAlpha.

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Minimizing a function involving a gamma function

In one of the proofs about some estimates of upper bounds of zeros of the Riemann zeta function I am studying, I have a function that looks like $V(\alpha, x, \delta,T)= \alpha e^{2\alpha(\frac{1}{T} +...
Josh's user avatar
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Expand product of sums in sympy by introducing new dummy indices

I want to use sympy to simplify some expressions which contain products of sums, this will require expanding out the products and cancelling equal terms. ...
cyfirx's user avatar
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Equating to zero as many of the highly linearly dependent entities as possible

I have $7$ variables $x_1$, ..., $x_7$ and a bunch of their linear combinations; specifically, denoting $s=x_1+...+x_7$, these combinations are $s-5(x_i+x_j)$, $1\leqslant i<j\leqslant7$ and $2s-5(...
მამუკა ჯიბლაძე's user avatar
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Line Integral in Octave

I want to compute Line Integral in Octave. There is a sample code that works fine in Matlab, but it does not work in Octave. ...
Julio Otuyama's user avatar
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20 views

Computationally evaluating messy symbolic sums involving geometric series

Let $t$ be a positive integer, let $p$ be a prime number, and let $q$ be a real number. I need to evaluate the sum $$ \sum_{\substack{1 \leq c \leq t \\ c \not \equiv 1 \pmod{p}}} q^{-\big((p-2)c + \...
Sebastian Monnet's user avatar
1 vote
1 answer
56 views

Magma - Differential field extension over a differential field

I want to construct the differential field $\mathbb{Q}(x,\,\log x,\,\log(\log x))$ in Magma. I have tried the following: ...
Mitchell Holt's user avatar
1 vote
0 answers
69 views

Hiper Calc app: symbolic integration step functions possible?

The app (android) Hiper Calc is rather powerful, with great CAS-capabilities. But it doesn't seem to have step functions, or Piecewise, or other ways to define conditional functions. I tried to mimic ...
Stef Pillaert's user avatar
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Meaning of Φ : Y → X

I am reading the book "Advances in Intelligent Systems and Computing" and I cannot understand a paragraph. Parametric approaches include the classical SOM and its probabilistic counterpart ...
tahasozgen's user avatar
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28 views

Given this function and one of its roots, is it possible to find a simpler expression that only has the original function's other two roots?

I have the function $$ \begin{align} f(x) = &\left(z × \cos(α) × (\cos(x + α - \tan(α)) + x × \sin(x + α - \tan(α))) - z × \cos\left(\frac{F}{z × \cos(α)}\right)\right)^2 \\ + &\left(z × \cos(...
Lawton's user avatar
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Interpreting/understanding the lambertW on Maple software

I decided to use the Maple software to help me solve dynamic optimization problems, and I found this final solution for T. (example in the picture). example What does the LambertW mean for the ...
Meg's user avatar
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Reduced form of symbolic expressions in sagemath

I am working in symbolic ring with two symbols: q and h. The relation between them is $q=e^h$. I want to do some algebra of matrices with symbolic entries. The entries of the input matrices consist of ...
Shruti's user avatar
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3 votes
2 answers
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Are there basic techniques for proving that no elementary antiderivative exists? [duplicate]

Background: It seems there is an algorithm called the full Risch algorithm that can decide whether a given function has an antiderivative that can be expressed in terms of elementary functions. There ...
WillG's user avatar
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Sympy does not recognise when two partial derivatives happen to be equal.

I am trying to find the first and second-order partial derivatives of a function of four variables $S$ using pythons symbolic math package sympy. The issue is that sympy does not automatically see ...
G-Shillcock's user avatar
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Evaluate $ \int\limits_0^{+\infty} x \tanh^{-1}\left(\frac{k x}{a}\right) \exp\left(-\frac{x^2}{b}\right) dx $ [closed]

I would like to evaluate these two integrals $$ I_1= \int\limits_0^{+\infty} x \tanh^{-1}\left(\frac{k x}{a}\right) \exp\left(-bx^2\right) dx$$ and $$ I_2=\displaystyle \int\limits_0^{+\infty} x^3 \...
Gallagher's user avatar
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Abel transform of Gauss Function and other bell shaped functions

Could you help me compute Abel transform of Gauss function. I need $$A_g[\sigma](x) = \int_x^\infty \frac{r}{\sqrt{r^2-x^2}} e^{-(\frac{r}{\sigma})^2} \, d\mathrm{r}, \,\,\,\,\,\, x\geq0, $$ where $\...
VojtaK's user avatar
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Comparing "diff" in Sagemath and Sympy

I am currently testing two programs of symbolic computation, one written in Sage and one in Sympy, which do similar computations. After running both of them, I found that the one written in Sympy is ...
Miguel Mars's user avatar
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2 votes
1 answer
428 views

Software for symbolic matrix operations

Is there a software for symbolic manipulation that can treat matrices as whole variables? For instance: solve A = B + C * A for A where ...
Max The Pax's user avatar
0 votes
2 answers
116 views

CAS software similar to Nspire on macOS 10.13 [closed]

Last edit (2023-01-26): This question (and answer) has been closed as off-topic, even though the What to ask here for this site says: Software that mathematicians use (except Mathematica, which has ...
dotnetCarpenter's user avatar
1 vote
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Expressing an equation in terms of its generators

I have a generating set of differential equations: $$ I_1=u_t\\ I_2=-uu_{tt}\\ I_3=u^{-1}\\ I_4=uu_{xxx}-6u^2u_x $$ where $u=u(x,t)$. and the coordinate is $(x,t,u,u_x,u_t,u_{tt},u_{xxx})$. Infact we ...
MB_18's user avatar
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1 vote
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135 views

Symbolic Evaluation of the Limit of an Undefined Function's Derivative in SymPy

Suppose that I have an unknown function $$f(x, y, z)$$ whose first order partial derivatives with respect to $(x, y, z$) are known to be everywhere smooth and well behaved (i.e., continuous, with no ...
Brendon Lucas's user avatar
2 votes
0 answers
27 views

How to get the conditions required to analyze the result? [closed]

I'm struggling to understand the method followed in the following analysis. Can someone please explain, how the author got the values of $Δ_1$ and $K_1$ and justify its analysis. I have tried to ...
Jack's user avatar
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Understanding Richardson's Theorem

Richardson's Theorem states that (quoting from Wolfram MathWorld): Let $\mathcal{R}$ be the class of expressions generated by The rational numbers and the two real numbers $\pi$ and $\ln 2$, The ...
a_guest's user avatar
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1 answer
181 views

Implementing a convolution integral symbolically using integration-by-parts

I'm working on implementing a convolution integral symbolically (not numerically), to be used in a mathematical modelling tool. This is more of a mathematical problem at this stage, NOT programming. ...
Ajay Menon's user avatar
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61 views

Is there a standard way of writing constructible numbers?

Every rational number can be represented by an irreducible fraction with a nonnegative denominator. For expressions involving integers, the four basic operations of arithmetic and square roots, you ...
Voekoevaka and Kixyy's user avatar
3 votes
1 answer
557 views

Software for solving symbolic matrix expressions

I've seen that there are multiple softwares to solve equations like sageMath, sympy, maxima... All of these seem to work great for scalars but I haven't found any software which can solve symbolic ...
jonithani123's user avatar
1 vote
0 answers
97 views

Computing inverse elements of symbolic matrices with binary variables

I'm working with symmetric, symbolic matrices $A$ with real coefficients and linear binary variables like $$ A = \begin{pmatrix} 0.5x_0 & 0.3x_1+0.002x_2 & 0 & 0 \\ 0....
Candlejack's user avatar
0 votes
2 answers
88 views

Derivative of complex quadratic form-functions with respect to a vector

I have a quadratic form like this: $Q_1=v_1^TC_1(x)v_1$ $Q_2=v_2^TC_2(y)v_2$ where $x,y,z$ - $3 \times 1$-vectors $J_1,J_2,C_1,C_2$ - $3 \times 3$-matrix $v_1=x-(J_1(x)y-z),v_2=x-(J_2(y)y-z)$ ...
dtn's user avatar
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3 answers
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Finding the symbolic expression for a particular matrix raised to the $n^{\text{th}}$ power

Consider the following matrix, with $a$ being a real number between $0$ and $1$. \begin{equation} \text{M}=\left[ \begin{array}{cc} 1-\frac{a^2}{3} & \frac{1}{3} \left(2 a-a^2\right) \\ \frac{2 ...
Stephen Strange's user avatar
1 vote
0 answers
58 views

sympy: group given expressions within other expressions, e.g. $a + 2 x y + b x + 2 x z$

I would like to use sympy to locate and group instances of a given symbolic expression within a set of expressions, so that the given expression may be replaced by a variable. For instance, with ...
Steve White's user avatar
2 votes
1 answer
109 views

Computing an integral depending on two parameters

A few days ago I stumbled upon this integral $$ f(y,c) :=\int_{-c-\sqrt{2}}^{-c+\sqrt{2}}\sqrt{2-(x+c)^2}\; \ln\left|\frac{x+y}{x-y}\right| \;\mathrm{d}x $$ where $c>0$ and $y\in(-c-\sqrt{2},-c+\...
Ludwig's user avatar
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1 vote
1 answer
33 views

Are there any symbolic solvers that can use matrices and vectors directly?

I frequently deal with matrix/vector calculus problems and would like to get the answers in matrix/vector form. The number of elements in these matrices and vectors is not specified, and the desired ...
Mastiff's user avatar
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1 answer
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How to apply Risch Algorithm by hand to solve integrals?

In a German forum, a user asked how the "Feynman"-trick works. The example was $$ f(x)=x e^{x} $$ Another user mentioned that the Risch algorithm should be taught. Therefore, I wonder ...
Elie Makdissi's user avatar
9 votes
0 answers
216 views

Composition of some linear differential operators $(D-A_n)...(D-A_1)$

Let $D=x\frac{d}{dx}$ and $A_i\in\mathbb{R}[[x]]$ for $i=1,...,n$. Let $B_i\in\mathbb{R}[[x]]$ for $i=1,...,n$ such that $$(D-A_n)...(D-A_1)=D^n+\sum_{i=1}^nB_iD^{n-i}.$$ Is there a well-known formula ...
Platonicsolids's user avatar
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37 views

Identifying Hamiltonian dynamics in 2D

While studying a physical system I've encountered an two dimensional autonomous ODE defined on a torus. The vector field describing it is a messy expression involving rational functions of ...
Radost's user avatar
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1 vote
1 answer
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Symbolic points in an elliptic curve over $\mathbb{Q}$ of the form $(u/e^{2},v/e^{3})$

Suppose I have an elliptic curve $E : y^{2} = x^{3} + D$ over $\mathbb{Q}$ , where $D$ is a symbolic constant (an integer). I want to define two points in the curve of the type $P = (u_{1}/e_{1}^{2}, ...
Atratrana Suna's user avatar
1 vote
1 answer
132 views

Symbolic point in an elliptic curve over $\mathbb{Q}$ in SageMath

I want to study the behavior of polynomials sitting in the co-ordinates of multiples of a rational point of an elliptic curve over $\mathbb{Q}, $ $E(\mathbb{Q}).$ Suppose if we take a point $P = (s,t) ...
Atratrana Suna's user avatar
2 votes
0 answers
811 views

How to use Octave with real numbers and syms at one time

I want to use GNU Octave to solve my equations. Equations contain syms variables and vectors with decimal numbers in them. How properly solve these equations without getting warnings that im doing ...
Tomáš Rompotl's user avatar
-1 votes
1 answer
181 views

Integral of Squared Sum of Sine [closed]

Let's say we have the following function $$ \int \left( \sum_{i = 1}^\infty a_i \sin(b_ix)\right)^2$$ Assuming that this series converges (it is actually a Fourier Series of the Lagrangian of the Wave ...
TheAkashain's user avatar
1 vote
1 answer
309 views

Algorithm for expressing Gröbner basis in terms of ideal generators

Given a polynomial ring and an ideal $$A \supset I = (f_1, ..., f_m)$$ there are plenty of implementations of an algorithm (e.g. Buchberger's) that produces a Gröbner basis $$G = (g_1, ..., g_n)$$ and ...
Somatic Custard's user avatar
0 votes
2 answers
87 views

How to prove a transcendental equation is never zero or sometimes zero?

I am an engineer by profession. Recently, while working on beams, I obtained the following transcendental equation: $$1-\cos(x)\cosh(x)+x^2\sin(x)\sinh(x)$$. I was wondering if this expression can ...
Thorongil's user avatar
0 votes
1 answer
52 views

How can you show that $(p \rightarrow q) \land (q \rightarrow r) \leftrightarrow (p \rightarrow r)$ is not a tautology?

Obviously, $(p \rightarrow q) \land (q \rightarrow r) \rightarrow (p \rightarrow r)$ due to Hypothetical Syllogism, but how about the converse? How can I disprove the statement?
Supakorn Srisawat's user avatar
2 votes
0 answers
53 views

Problem evaluating Wolfram Symbolic Integration Result

****** Note: In response to the comment below, I've modified the problem ****** I'm trying to evaluate the following indefinite integral $$ \Large\int x^{-\frac{3}{2}} e^{-\frac{K}{x}\Big[(x-a)^2+(x-b)...
Sharat V Chandrasekhar's user avatar
5 votes
2 answers
268 views

Handling elementary but messy computations in proofs

Often when working on a proof, I get to a computation which appears to be elementary (e.g. requiring only standard algebra and perhaps calculus) but messy. Solving this via pen and paper is tedious ...
SRobertJames's user avatar
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2 votes
1 answer
72 views

Why does the function $\frac{1}{x^{2222}}$ seem to have *two* asymptotes at around $\pm0.744$?

When screwing around with the Dirac-delta function in Desmos i noticed that when i multiply it by a really low-degree rational function $$f(x)=\frac{1}{x^i}$$ , the resulting function has the domain $$...
Lord Commander's user avatar
0 votes
3 answers
44 views

How to write that the only integers that have a multiplicative inverse are $1$ and $-1$, in symbolic form

Could it be something like for $(1,-1) \in \mathbb{Z}$, there exists $y \in \mathbb{Z}$, such that $xy = 1$?
Anonymous Molecule's user avatar
1 vote
0 answers
46 views

what is the meaning of this element $\varphi(f|_Y)?$

I have some confusion in symbolics . My confusion is given below marked in red colour what is the meaning of this element $\varphi(f|_Y)?$ My main confusion about the symbolics is $(f|_Y)$?
jasmine's user avatar
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1 vote
0 answers
55 views

What is this hybrid symbolic-numeric algorithm for solving non-linear ODEs called?

I'm an engineer so please forgive my lack of knowledge in exact mathematical terminology: For linear differential equations, it is possible to use the principle of superposition or convolution ...
MathX's user avatar
  • 153
1 vote
1 answer
325 views

Factor SYMBOLIC third degree polynomial

I apology in advance for the long post... Say I have a very general 3rd-degree polynomial $$x^3 + a_1x^2 + a_2x+a_3$$ I would really like to factor that polynomial as a product of a 1st order and a ...
Anthony A.'s user avatar
1 vote
2 answers
71 views

Finding $b$ such that $y=mx+c$ is tangential to $y=b^x$.

I was thinking about the following problem: Find $b$ such that $y=mx+c$ is tangential to $y=b^x$. My work so far: To satisfy the condition we need to find $x$ such that $\frac d{dx}\left[b^x\right]=\...
Kyan Cheung's user avatar
  • 3,194
3 votes
1 answer
155 views

How to find solve for second order pde with initial conditions using Wolfram Mathematica?

I have next task: $$ \frac{\partial^2 u}{\partial x \partial y} = 0,~ u(x,x^2) = 0,~ \frac{\partial u}{\partial x}(x, x^2) = \sqrt{|x|},~|x| < 1 $$ I write this, but it don't work: ...
Wythuk's user avatar
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