The art of manipulating an algebraic expression into the desired form.
26
votes
6answers
2k views
Finding real roots of negative numbers (for example, $\sqrt[3]{-8}$)
Say I want to quickly calculate $\sqrt[3]{-8}$, to which the most obvious solution is $-2$.
When I input $\sqrt[3]{-8}$ or Power[-8, 3^-1], Mathematica gives the ...
16
votes
4answers
415 views
How do I introduce a new variable in a trigonometric equation?
I have the trigonometric equation
\begin{equation*}
\sin^8 x + 2\cos^8 x -\dfrac{1}{2}\cos^2 2x + 4\sin^2 x= 0.
\end{equation*}
By putting $t = \cos 2x$, I have
\begin{equation*}
\dfrac{3}{16} t^4+ ...
16
votes
5answers
484 views
How do I get my equation to have the form $(x-a)^2 + (y-b)^2 + (z-c)^2-d = 0$?
I want Mathematica to express the equation $$-11 - 2 x + x^2 - 4 y + y^2 - 6 z + z^2=0$$ in the form $$(x - 1)^2 + (y - 2)^2 + (z - 3)^2 - 25=0$$ How do I tell Mathematica to do that?
14
votes
2answers
294 views
How to define a non-standard algebra in Mathematica?
I want to define an algebra, where there are three elements: 0, 1 and $\infty$ and two operations, addition and multiplication defined, both commutative:
$$\begin{align*}
0+0&=0\\
0+1&=1\\
...
11
votes
4answers
788 views
Is it possible to have Mathematica move all terms to one side of an equation?
I have an inequality expression that I would like to express in terms of the relation of the parameters to zero. More simply, I want to have mathematica move all the terms to one side of the ...
11
votes
4answers
407 views
Is there a way to Collect[] for more than one symbol?
Oftentimes you find yourself looking for polynomials in multiple variables. Consider the following expression:
a(x - y)^3 + b(x - y) + c(x - y) + d
as you can ...
10
votes
6answers
900 views
Replacing composite variables by a single variable
To replace a single variable by another variable, one can simply use the the replace all (/.) operator (e.g., ...
10
votes
2answers
967 views
Why doesn't Mathematica expand Cos[x]^3 Sin[x]^2?
I found some examples of Mathematica's commands usage in an old manual but the program gives me different result than expected
...
9
votes
1answer
151 views
Why is ReplaceAll behaving like this?
I'm learning to use the ReplaceAll function and I found the behavior of which is quite confusing.
For
Sqrt[f[x, y]] /. f[___] -> u
Mathematica returns
...
9
votes
2answers
474 views
Expand modulus squared
Is it possible to make a function in Mathematica that expands expressions of the form
$$|z + w|^2 = |z|^2 + 2\text{Re} \overline{z}w + |w|^2?$$
Preferably it should also be able to handle things ...
9
votes
3answers
257 views
InverseSeries of multiple variables and multiple equations
CONTEXT
Let us consider a bit of the Universe in which we draw spheres
(see a high resolution image here). Astronomers have shown that the density within these spheres
could be predicted quite ...
8
votes
3answers
329 views
How can I convert a complex number a+b I to the exponent form A Exp(I phi)?
When I have an expression such as:
(1/4 + I/4) ((1 - 2 I) x + Sqrt[3] y)
it is hard to get an intuition of the number.
So I want to convert it to the complex ...
8
votes
3answers
475 views
How can I convert x^2 to x*x?
When I try the following code:
a b^2 c /. b c -> e
Mathematica gives me:
a b^2 c
but what I want is:
...
8
votes
4answers
299 views
“Evaluating” polynomials of functions (Symbols)
I want to implement the following type evaluation symbolically
$$(f^2g + fg + g)(x) \to f(x)^2 g(x) + f(x) g(x) + g(x)$$
In general, on left hand side there is a polynomial in an arbitrary number of ...
8
votes
2answers
261 views
Shaping/simplifying equations in a certain way
A problem I am occasionally facing is to simplify an equation not to it's shortest form but to a form that is simple by other means. Often, this is grouping the term according to certain functions, ...
8
votes
1answer
193 views
Most efficient way to determine conclusively whether an algebraic number is zero
Let x be an algebraic number of unspecified degree, expressed using arithmetic, rational powers, and algebraic integers (edit: ...
7
votes
6answers
761 views
How do I replace a variable in a polynomial?
How do I substitue z^2->x in the following polynomial z^4+z^2+4?
z^4+z^2+4 /. z^2->x
...
7
votes
3answers
253 views
Distances between points in periodic cube
How can one implement more efficiently/elegantly/memory savvily the following function which returns a matrix of all Euclidian distances between points in 3D within a cube of width ...
7
votes
4answers
358 views
How to implement dual numbers in Mathematica?
I wonder how can I implement dual numbers in Mathematica, so that all functions work well with them (as with complex numbers).
Particularly, for each function $f$, ...
7
votes
1answer
253 views
How do I expand a sum?
I have a problem with Mathematica's symbolic manipulations. As an example, consider the following expression:
$$\sum _{i=1}^n -2 x_i \left(-a x_i-b+y_i\right)=0$$
How do I get Mathematica to expand ...
7
votes
1answer
71 views
ToNumberField won't recognize Root[…] as explicit algebraic number
In Mathematica 9.0.1, it appears that ToNumberField will not always recognize a Root object as an explicit algebraic number.
...
7
votes
2answers
629 views
How can I rationalize the denominator of an expression?
Mathematica doesn't rationalize the denominator automatically, and I haven't found anything in the documentation about it. But I found an old post on MathGroup, which proposes a solution using ...
7
votes
2answers
293 views
Manipulating an equation into standard quadratic form?
Say I have an equation of the form
$$u s + \frac{1}{v} + \frac{1}{p s + q} = 0$$
(or any form that can be written as a standard quadratic, really, the above form is just an example; they'll all be ...
7
votes
1answer
187 views
RootSum result manipulation/simplification
Consider the sum
sum1 = Sum[ k/( k^7 - 2 k + 3), {k, Infinity}]
...
6
votes
4answers
854 views
Checking if two trigonometric expressions are equal
Say I have two trigonometric expressions which are a bit complicated. Is there a quick way to check if they reduce to the same thing (that they are equal) using Mathematica?
I was solving this: $y'' ...
6
votes
4answers
2k views
Factoring polynomials to factors involving complex coefficients
I've run into some problems using Factor on polynomials with complex coefficient factors. Reading the documentation it looks like it only factors over the ...
6
votes
3answers
115 views
Is there any way to collect only variables with a specific power?
Suppose I've got this:
In[13]:= Expand[(a + b) (b + c) (c + a)]
Out[13]= a^2 b + a b^2 + a^2 c + 2 a b c + b^2 c + a c^2 + b c^2
And I want to collect only ...
6
votes
0answers
57 views
Apart may use Padé method: what's that?
How does Apart work? The page tutorial/SomeNotesOnInternalImplementation#7441 says, "Apart ...
5
votes
3answers
460 views
What function can I use to evaluate $(x+y)^2$ to $x^2 + 2xy + y^2$?
What function can I use to evaluate $(x+y)^2$ to $x^2 + 2xy + y^2$?
I want to evaluate It and I've tried to use the most obvious way: simply typing and evaluating $(x+y)^2$, But it gives me only ...
5
votes
3answers
366 views
Square both sides of an equation?
Can I define an equation (for example, x+1 == y^2 + 2), and tell Mathematica to square both sides?
If not, what is an equivalent way to achieve this?
5
votes
6answers
304 views
How to simplify a complicated Sum in terms of power Sums?
For example, I have:
$a=\sum _{r=1}^n x_r
\left(\left(\sum _{i=1}^n
x_i-x_r\right){}^2-\sum
_{i=1}^n x_i^2\right)$
...
5
votes
2answers
108 views
How to protect pattern or subexpression when distributing / expanding expression?
I've got an expression like
expr = (1-x)(a+b)
that I would like to distribute / expand while keeping factors of (1-x) intact, ...
5
votes
4answers
511 views
How to get exact roots of this polynomial?
The equation $$ 64x^7 -112x^5 -8x^4 +56x^3 +8x^2 -7x - 1 = 0 $$
has seven solutions $x = 1$, $x = -\dfrac{1}{2}$ and $x = \cos \dfrac{2n\pi}{11}$, where $n$ runs from $1$ to $5$. With ...
5
votes
1answer
145 views
Numbered symbols
I work with an exterior algebra over $R^n$. I have the basis $\{1,\omega_i\}_{i=1}^n$ in this algebra, and my differential operator is defined as
$$d\omega_k=\sum_{i>j>0,i+j=k} (i-j)w_i\wedge ...
5
votes
1answer
75 views
How to apply tags to expression terms?
I often see on this site and at the mathgroup the repeated questions on how to rearrange expression that Mathematica "likes" to keep in one form, but the user prefers in another. Consider this trivial ...
4
votes
1answer
438 views
Reduce an equation by putting a new variable
I have the following equation given:
$$ (26-x)\cdot\sqrt{5x-1} -(13x+14)\cdot\sqrt{5-2x} + 12\sqrt{(5x-1)\cdot(5-2x) }= 18x+32. $$
In order to solve it, I want to substitute $t = \sqrt{5x - ...
4
votes
2answers
278 views
expanding a polynomial and collecting coefficients
I'm trying to expand the following polynomial
...
4
votes
1answer
223 views
Move variable to one side of the equation
Say if I have a formula like so:
a1*a2*a3^(a4 + 1)*(1 - E^(a5*a6/a3^a4/a2)) == 0
How do I move a3 to the right? I've tried to follow other examples here on stack ...
4
votes
1answer
162 views
Finding mappings between expressions
Suppose we have an expression of the form:
$j=\frac{A\left(t\right)}{B\left(t\right)}=\frac{C\left(s\right)}{D\left(s\right)}$
That is, $j$ can be expressed either as a function of $t$, or as a ...
3
votes
7answers
215 views
Defining a function that completes the square given a quadratic polynomial expression
How can I write a function that would complete the square in a quadratic polynomial expression such that, for example,
CompleteTheSquare[5 x^2 + 27 x - 5, x]
...
3
votes
2answers
1k views
How to convert a system of parametric equations to a normal equation?
For example, I have a system of parametric equations (R is a constant number) :
...
3
votes
2answers
102 views
Unexpected side effect of removing the Orderless attribute from Times
First I make Times orderless:
ClearAttributes[Times, Orderless];
Then I evaluate
...
3
votes
3answers
211 views
Is it possible to use Composition for polynomial composition?
I want to do this:
$P = (x^3+x)$
$Q = (x^2+1)$
$P \circ Q = P \circ (x^2+1) = (x^2+1)^3+(x^2+1) = x^6+3x^4+4x^2+2$
I used Composition for testing if that could ...
3
votes
3answers
238 views
Limiting form of a polynomial expression
When simplifying an expression by hand, a trick that is often used is to remove terms that are lower powers of the independent variable, for instance, as $x \rightarrow \infty$,
$x^2 + x$
becomes ...
3
votes
5answers
295 views
How to group certain symbolic expressions?
For example, I have the following expression :
A( 2 x1 + B(y1 + y2) + 2 x2 )
How do I make the output look like this (grouping ...
3
votes
3answers
125 views
How to extract phase angle from sinusoid
I'm doing some electric circuit calcualtions and I'm trying to get the phasor representation of some arbitrary function of Sin or Cos. Could be complex like:
...
3
votes
2answers
95 views
3
votes
1answer
40 views
Non commutative multiply- expand expression
I began to use Mathematica a few days. My problem is: How to expand expression like $(a+b)*(a+b)$, where this multiplication is non commutative? Mathematica can do this?
3
votes
2answers
152 views
Eliminate several variables between five Conic_section equations
I want to eliminate x1, x2, y1, y2 between these 5 equations.
I tried
...
3
votes
1answer
157 views
How can I express an algebraic expression as a product?
I have a simply expression that I can see that I can express as a product.
...
