gfun[indicialpolynomial] Indicial polynomial of a linear differential equation at a point
Calling Sequences
indicialpolynomial(deq, y(z), pt)
indicialpolynomial(dop, [Dz,z], pt)
indicialpolynomial(listcoeffs, z, pt)
Parameters
deq - a linear differential equation with polynomial coefficients with or without initial conditions;
dop - a linear differential operator in Dz, with coefficients that are polynomials in z
[Dz,z] - the corresponding variables;
listcoeffs - a list of polynomials in z representing the coefficients of a linear differential equation
z - the corresponding variable
pt - either infinity or a rational number or a RootOf of an irreducible polynomial
Description
- This command computes the indicial polynomial as DEtools[indicialeq], with a different syntax and sometimes much faster.
- This command is part of the
gfunpackage, so it can be used in the formindicialpolynomial(..)only after executing the commandwith(gfun). However, it can always be accessed through the long form of the command by usinggfun[indicialpolynomial](..).
Examples
> with(gfun):> f:=exp(z):> deq:=holexprtodiffeq(f,y(z));\[\{\frac{d}{d z}y \! \left(z \right)-y \! \left(z \right), y \! \left(0\right) = 1\}\]
> indicialpolynomial(deq,y(z),infinity);\[1\]
> indicialpolynomial(deq,y(z),0);\[z\]
> deq:={(20*z^6+12*z^5)*y(z)+(4*z^7+z^2+3*z-9)*diff(y(z),z)+(z^3-3*z)*diff(diff(y(z)
,z),z), y(0) = 1};\[\{\left(20 z^{6}+12 z^{5}\right) y \! \left(z \right)+\left(4 z^{7}+z^{2}+3 z -9\right) \left(\frac{d}{d z}y \! \left(z \right)\right)+\left(z^{3}-3 z \right) \left(\frac{d^{2}}{d z^{2}}y \! \left(z \right)\right), y \! \left(0\right) = 1\}\]
> indicialpolynomial(deq,y(z),RootOf(z^2-3));\[z^{2}+\left(\frac{37 \textrm{RootOf}\left(\_Z^{2}-3\right)}{2}-2\right) z\]
> dop:=(2*z-2)*Dz-1;\[\left(2 z -2\right) \mathit{Dz} -1\]
> indicialpolynomial(dop,[Dz,z],1);\[z -\frac{1}{2}\]
> L:=[5,72*z-42,36*z^2-36];\[[5, 72 z -42, 36 z^{2}-36]\]
> indicialpolynomial(L,z,1);\[-\frac{7}{12} z +z^{2}\]
>