ElementaryFunctionStructurePackage(R, F)ΒΆ

efstruc.spad line 78

ElementaryFunctionStructurePackage provides functions to test the algebraic independence of various elementary functions, using the Risch structure theorem (real and complex versions). It also provides transformations on elementary functions which are not considered simplifications.

irootDep: Kernel F -> Union(vec: Vector Fraction Integer, func: F, fail: Boolean)
irootDep(k) is a local function with a conditional implementation.
normalize: (F, Symbol) -> F
normalize(f, x) is normalize([f], [x])
normalize: (List F, List Symbol) -> List F
normalize([f1, ..., fn], lx) rewrites f1, ..., fn using the least possible number of real algebraically independent kernels. Additionally, it tries to ensure that any expression in resulting kernels which syntactically depends on a variable x from lx has nonzero derivative with respect to x.
normalize: F -> F
normalize(f) rewrites f using the least possible number of real algebraically independent kernels.
normalize: List F -> List F
normalize(lf) is normalize(lf, [])
realElementary: (F, Symbol) -> F
realElementary(f, x) rewrites the kernels of f involving x in terms of the 4 fundamental real transcendental elementary functions: log, exp, tan, atan.
realElementary: F -> F
realElementary(f) rewrites f in terms of the 4 fundamental real transcendental elementary functions: log, exp, tan, atan.
realLiouvillian: (F, Symbol) -> F
realLiouvillian(f, x) rewrites f elementary kernels of f in terms 4 fundamental real elementary functions: log, exp, tan, atan. Additionally, it rewrites Liouvillian functions of x as indefinite integrals to support better normalization.
realLiouvillian: F -> F
realLiouvillian(f) rewrites f elementary kernels of f in terms 4 fundamental real elementary functions: log, exp, tan, atan. Additionally, it rewrites Liouvillian functions as indefinite integrals to support better normalization.
rischNormalize: (F, List Symbol) -> Record(func: F, kers: List Kernel F, vals: List F)
rischNormalize(f, lx) is rischNormalize([f], lx)
rischNormalize: (F, Symbol) -> Record(func: F, kers: List Kernel F, vals: List F)
rischNormalize(f, x) is rischNormalize([f], [x]) rischNormalize(f, x) returns [g, [k1, ..., kn], [h1, ..., hn]] such that g = normalize(f, x) and each ki was rewritten as hi during the normalization.
rischNormalize: (List F, List Symbol) -> Record(funcs: List F, kers: List Kernel F, vals: List F)
rischNormalize(lf, lx) returns [lg, [k1, ..., kn], [h1, ..., hn]] such that lg = normalize(lf, lx) and each ki was rewritten as hi during the normalization.
rmap: (Kernel F -> F, F) -> F
rmap(f, e) rewrites e replacing each kernel k in e by f(k)
rootNormalize: (F, Kernel F) -> F
rootNormalize(f, k) returns f rewriting either k which must be an nth-root in terms of radicals already in f, or some radicals in f in terms of k.
tanQ: (Fraction Integer, F) -> F
tanQ(q, a) is a local function with a conditional implementation.
validExponential: (List Kernel F, F, Symbol) -> Union(F, failed)
validExponential([k1, ..., kn], f, x) returns g if exp(f)=g and g involves only k1...kn, and “failed” otherwise.