gfun[algebraicsubs] - substitute an algebraic function into an holonomic one

Calling Sequence

algebraicsubs(deq, f(t), P, y, h(x), <branch>, <homogeneous=bool>, <operator=name>)

algebraicsubs(deq, eq, y(z), <ini>, <iniconds>, <homogeneous=bool>, <operator=name>)

Parameters

deq - linear differential equation for the outer function f(t), with polynomial coefficients

P - polynomial in the algebraic-curve indeterminates x and y

y - name of the algebraic-curve dependent indeterminate

h(x) - name and variable of the resulting composition

eq, y(z), ini, iniconds - parameters of the supported historical calling sequence

ini - (optional) initial conditions to specify a solution of eq

homogeneous - (optional) boolean requesting a homogeneous output equation; the default is false

operator - (optional) name of the differential operator

In the preferred calling sequence, deq is an equation for the outer function f(t), P is a polynomial in the independent indeterminates x and y, and the output is an equation for h(x)=f(y(x)). The optional set branch specifies initial conditions for the algebraic branch, such as {y(0)=1}. The historical calling sequence remains supported.

Description

Let f be the holonomic function defined by the equation deq, and g be the algebraic equation defined by eq, then gfun[algebraicsubs] outputs a differential equation satisfied by the composition f@g, which is holonomic by closure properties of holonomic functions.
With homogeneous=true, all powers of the algebraic inner function, including its constant power, are retained as matrix columns. This computes a homogeneous dependence directly.
With operator=name, the supplied name is used verbatim and initial conditions are not computed. A homogeneous relation is returned as a polynomial in that name; an inhomogeneous relation q+L(y)=0 is returned as L(name)=-q.
Let d1 be the differential order of deq, and d2 be the degree of eq. If the equation deq is homogeneous, then the order of f@g is at most d1*d2 otherwise it is at most (d1+1)*d2
When initial conditions are requested, they are returned only if the supplied initial conditions for eq determine a unique local Puiseux branch and the selected composition has an ordinary Taylor expansion at the origin. If no branch, several branches, or only a common polynomial prefix is determined, or if the composition has fractional powers at the origin, only the differential equation is returned.

Examples

The differential equation satisfied by cos(t):

> with(gfun):
deq := (D@@2)(f)(t)+f(t):

The algebraic equation satisfied by sqrt(1-4*t):

> eq := algfuntoalgeq(sqrt(1-4*t),f(t)):

The differential equation satisfied by cos(sqrt(1-4*t)):

> algebraicsubs(deq,eq,f(t));

\[-4 f \! \left(t \right)+2 \frac{d}{d t}f \! \left(t \right)+\left(-1+4 t \right) \left(\frac{d^{2}}{d t^{2}}f \! \left(t \right)\right)\]

gfun[algebraicsubs] - substitute an algebraic function into an holonomic one

Calling Sequence

Parameters

Description

Examples

See Also