How to expand taylor series/polynomials about Q=0 , and then extract coefficients as a list
example :
taylor ( (sin(q)), q, 0, 9); //taylor expansion for first 9 terms gives the next line
(%o1)/T/ q\-q^3/6+q^5/120\-q^7/5040+q^9/362880+...
then using coeff ((%o1), q ^n); gives me the coefficient at n only, what i want is a list for all the coefficients of that expression
Try coeff plus makelist, e.g. something like: makelist(coeff(%o1, q, n), n, 0, 9);
Edit:
I see now that I misread your question and there is already an answer. Nevertheless I will keep it because it is related to your question.
Use powerseries instead of taylor:
(%i1) expr:powerseries(sin(x),x,0);
inf
==== i2 2 i2 + 1
\ (- 1) x
(%o1) > -----------------
/ (2 i2 + 1)!
====
i2 = 0
You can access the coefficient by the args or part function
(%i2) op(expr);
(%o2) sum
(%i3) args(expr);
i2 2 i2 + 1
(- 1) x
(%o3) [-----------------, i2, 0, inf]
(2 i2 + 1)!
(%i4) part(expr,1);
i2 2 i2 + 1
(- 1) x
(%o4) -----------------
(2 i2 + 1)!
(%i5) args(expr)[1];
i2 2 i2 + 1
(- 1) x
(%o5) -----------------
(2 i2 + 1)!
If you want to change the index variable:
(%i6) niceindices(expr),niceindicespref=[n];
inf
==== n 2 n + 1
\ (- 1) x
(%o6) > ---------------
/ (2 n + 1)!
====
n = 0
Related
Here's the initial premise: two sums s1 and s2 are added; the sum element expressions have a common factor a[n].
s1: sum(r1[m,q]*b[m,n]*a[n],n,0,N)$
s2: sum(r2[m,q]*c[m,n]*a[n],n,0,N)$
s1+s2;
I expect the sums to be combined and the common element expression a[n] factored out:
s12: sum(a[n]*(r1[m,q]*b[m,n]+r2[m,q]*c[m,n]),n,0,N);
However, I'm unable to make Maxima produce such contraction. The most simplification I was able to obtain was using sumcontract(s1+s2) and it results in two sums without the common element being factored out:
r1[m,q]*sum(b[m,n]*a[n], n,0,N) + r2[m,q]*sum(c[m,n]*a[n], n,0,N);
How to make Maxima produce the factored out expression from s1+s2 as in s12 above?
NOTE: If we remove the r1 and r2, then the factor(sumcontract(s1+s2)) indeed results in the expected s12 expression. However, with both present, it results in two sums and does not factor out the a[n] as mentioned.
How about this. I've applied sumcontract, intosum, and factor.
(%i1) s1: sum(r1[m,q]*b[m,n]*a[n],n,0,N)$
(%i2) s2: sum(r2[m,q]*c[m,n]*a[n],n,0,N)$
(%i3) s1 + s2;
N N
==== ====
\ \
(%o3) r2 > c a + r1 > b a
m, q / m, n n m, q / m, n n
==== ====
n = 0 n = 0
(%i4) intosum (%);
N N
==== ====
\ \
(%o4) > c r2 a + > b r1 a
/ m, n m, q n / m, n m, q n
==== ====
n = 0 n = 0
(%i5) sumcontract (%);
N
====
\
(%o5) > (c r2 a + b r1 a )
/ m, n m, q n m, n m, q n
====
n = 0
(%i6) factor (%);
N
====
\
(%o6) > (c r2 + b r1 ) a
/ m, n m, q m, n m, q n
====
n = 0
In this, intosum is pushing constant factors back into the sum.
I have a function in Maxima I am differentiating then attempting to find the value at which this is zero. When I use solve(), however, I am not given a solution. Why is this, and how can I work around it?
(%i1) f(x):=(-5*(x^4+5*x^3-3*x))/(x^2+1);
(%o1) f(x):=((-5)*(x^4+5*x^3+(-3)*x))/(x^2+1)
(%i2) df(x):=''(diff(f(x), x));
(%o2) df(x):=(10*x*(x^4+5*x^3-3*x))/(x^2+1)^2-(5*(4*x^3+15*x^2-3))/(x^2+1)
(%i3) solve(df(x), x);
(%o3) [0=2*x^5+5*x^4+4*x^3+18*x^2-3]
The function solve is not too strong; there are many problems it can't solve. A stronger version is under development. In the meantime, try the add-on package to_poly_solve. Here's what I get:
(%i1) df(x) := (10*x*(x^4+5*x^3-3*x))/(x^2+1)^2-(5*(4*x^3+15*x^2-3))/(x^2+1) $
(%i2) load (to_poly_solve) $
(%i3) to_poly_solve (df(x), x);
(%o3) %union([x = - 2.872468527640942], [x = - 0.4194144025323134],
[x = 0.3836388367122223], [x = 0.2041221431132173 - 1.789901606296292 %i],
[x = 1.789901606296292 %i + 0.2041221431132173])
Something which is maybe a little surprising is that to_poly_solve has returned a numerical solution instead of exact or symbolic. Tracing allroots shows that to_poly_solve has constructed a quintic equation and punted it to allroots. Since the general quintic doesn't have a solution in terms of radicals, and even in special cases it's probably very messy, maybe it's most useful to have a numerical solution anyway.
Try plot2d(df(x), [x, -3, 1]) to visualize the real roots returned above.
You can try to find a numerical solution. I don't know why solve does not try this. Either you take the ouput of aolveor you do hte folölowing:
(%i1) f(x):=(-5*(x^4+5*x^3-3*x))/(x^2+1);
4 3
(- 5) (x + 5 x + (- 3) x)
(%o1) f(x) := ---------------------------
2
x + 1
(%i2) df(x):=''(diff(f(x), x));
4 3 3 2
10 x (x + 5 x - 3 x) 5 (4 x + 15 x - 3)
(%o2) df(x) := ---------------------- - --------------------
2 2 2
(x + 1) x + 1
Bring it to a common denominator and extract the numerator:
(%i3) xthru(df(x));
4 3 2 3 2
10 x (x + 5 x - 3 x) - 5 (x + 1) (4 x + 15 x - 3)
(%o3) ------------------------------------------------------
2 2
(x + 1)
(%i4) num(%);
4 3 2 3 2
(%o4) 10 x (x + 5 x - 3 x) - 5 (x + 1) (4 x + 15 x - 3)
use allsrootsto find the roots of a polynomial numerically
(%i5) allroots(%);
(%o5) [x = 0.3836388391066617, x = - 0.4194143906217701,
x = 1.789901606296292 %i + 0.2041221431132174,
x = 0.2041221431132174 - 1.789901606296292 %i, x = - 2.872468734711326]
skip the complex solutions
(%i6) sublist(%,lambda([t],imagpart(rhs(t))=0))
;
(%o6) [x = 0.3836388391066617, x = - 0.4194143906217701,
x = - 2.872468734711326]
newbie Maxima question
I have a transfer function in Maxima
E1 : y = K_i*s/(s^2 + w^2);
I'd like to have the closed-form of the equation affter applying the bilinear transform
E2 : s = (2/Ts*(z-1)/(z+1));
I would like to get the transfer function for z, by substituing s by equation E2. How should I proceed?
Regards
Note that subst can apply one or more substitutions stated as equations. In this case, try subst(E2, E1).
That will probably create a messy result -- you can simplify it somewhat by applying ratsimp to the result.
Here's what I get from that.
(%i2) E1 : y = K_i*s/(s^2 + w^2);
K_i s
(%o2) y = -------
2 2
w + s
(%i3) E2 : s = (2/Ts*(z-1)/(z+1));
2 (z - 1)
(%o3) s = ----------
Ts (z + 1)
(%i4) subst (E2, E1);
2 K_i (z - 1)
(%o4) y = ------------------------------
2
4 (z - 1) 2
Ts (z + 1) (------------ + w )
2 2
Ts (z + 1)
(%i5) ratsimp (%);
2
2 K_i Ts z - 2 K_i Ts
(%o5) y = -----------------------------------------------
2 2 2 2 2 2 2
(Ts w + 4) z + (2 Ts w - 8) z + Ts w + 4
I have an matrix in Maxima, let´s say (for simplification of the problem):
A: matrix([2*(a^2+b^2+c^2)])
But I know that:
a^2+b^2+c^2 = 1
How do I simplify that matrix in Maxima in terms of that equation, in order to obtain A = [2]?
I found the solution:
A: matrix([2*(a^2+b^2+c^2)]);
eq: a^2+b^2+c^2 = 1;
scsimp(A, eq);
You can use tellrat.
(%i1) A:matrix([2*(a^2+b^2+c^2)])
(%o1) [ 2 2 2 ]
[ 2 (c + b + a ) ]
(%i2) a^2+b^2+c^2 = 1
2 2 2
(%o2) c + b + a = 1
(%i3) solve(%,a^2)
2 2 2
(%o3) [a = (- c ) - b + 1]
(%i4) tellrat(%[1])
2 2 2
(%o4) [c + b + a - 1]
(%i5) algebraic:true
(%o5) true
(%i6) rat(A)
(%o6)/R/ [ 2 ]
(%i7) untellrat(a)
(%o7) []
I have an expression:
(b+2*ab+a+1)/c
I want to use Maxima to factor the equation treating (b+1) as a factor.
i.e. I want the expression in the following form:
[(b+1)(1+a)+ab]/c
Any help would be appreciated.
Well, my advice is first isolate the numerator, then get the quotient and remainder after dividing by b + 1, then put the pieces back together.
(%i1) display2d : false $
(%i2) expr : (b + 2*a*b + a + 1)/c $
(%i3) num (expr);
(%o3) 2*a*b+b+a+1
(%i4) divide (num (expr), b + 1);
(%o4) [2*a+1,-a]
(%i5) first(%o4) * (b + 1) + second(%o4);
(%o5) (2*a+1)*(b+1)-a
(%i6) (first(%o4) * (b + 1) + second(%o4)) / denom (expr);
(%o6) ((2*a+1)*(b+1)-a)/c
(%i7) is (equal (%o6, expr));
(%o7) true
Note that divide returns two values; first is the quotient and second is the remainder.