Function Repository Resource:

PowerAffineExpand

Compute branch-correct expansions of power for univariate polynomials and rational functions with real coefficients

Contributed by: Jayanta Phadikar

ResourceFunction["PowerAffineExpand"][p, n, x]

expands p^n for the real-valued rational function p in x as a sum of powers of affine factors.

Details

ResourceFunction["PowerAffineExpand"] gives results that are always correct under the assumption that the argument x is real-valued.

The argument p need not have a denominator, that is, it can be a univariate polynomial.

An affine factor is a degree-one polynomial of the form ax+b with a!=0. In this function, we use the normalized affine form 1+αx.

An affine power expansion is a representation of power of a polynomial or rational function that replaces the original expression by a constant times a product of powers of normalized affine factors, together with a piecewise integer multiple of 2π ⅈ so that the result coincides exactly with the principal branch value of the original expression on the real line.

The coefficients must be real-valued and numeric. Inputs with complex or symbolic coefficients are left unevaluated.

The correctness of the expansion using PowerAffineExpand is guaranteed for real x only, where branch jumps occur only when real affine factors become negative. For complex x, the current implementation may miss 2π ⅈ adjustments when non-real affine factors map x onto the negative real axis; in such cases the result can differ from Power[expr, n] by integer multiples of 2π ⅈ.

ResourceFunction["PowerAffineExpand"][Power[p,n],x] is equivalent to PowerAffineExpand[p, n, x]. This convenience form exists for pipeline symmetry with PowerExpand and ResourceFunction["LogAffineExpand"].

Examples

Basic Examples (3)

Expand a simple quadratic polynomial:

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Expand a simple rational function:

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Use the equivalent two arguments form:

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Scope (5)

Complex-conjugate factorization of a real quadratic:

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Verify that the expansion correctly represents the original function in the real line:

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$Chop@N@Table[(x^2 + x + 1)^(3/4) - res, {x, -100, 100}]$

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Branch-corrected expansion near a real root:

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On the principal branch, when x>2 the factor 1-x/2 is negative real. PowerAffineExpand automatically inserts a compensating piecewise constant term -2π ⅈ/3 so the result is exactly equal to the original input on the principal branch. A similar branch correction for a simple rational function:

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For a rational input, the expansion keeps the numerator's affine factors in the numerator and the denominator's as inverse powers, with necessary piecewise branch/sign corrections:

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Verify that the result is correct:

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$Plot[Chop[((x^2 - 1)/(x^2))^(1/2) - res], {x, -100, 100}]$

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Root objects may appear in the result:

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Use ToRadicals to express using radicals:

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Root objects in the expansion of a rational function:

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Properties and Relations (3)

PowerExpand applies identities like Power(a b, n)=Power(a, n)*Power(b, n) under generic assumptions and without managing branches, so it may stop at non-affine factors and either omit necessary 2π ⅈ corrections or emit complex corrections:

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PowerAffineExpand always expands into affine factors and simplified 2π ⅈ corrections:

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Similarly for rational expressions:

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Possible Issues (2)

For PowerAffineExpand to work all constants appearing in the expression should be real and numeric. It won't work for symbolic or complex parameters:

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PowerAffineExpand gives an expansion which is guaranteed to be correct for real x only, and may not be valid for complex x:

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Neat Examples (4)

Expansion of a high degree polynomial:

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Build a polynomial with six factors. PowerAffineExpand reduces its power to an expression with one affine power per factor, inserting a piecewise 2π ⅈ term for principal-branch correctness:

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For real quadratics with negative discriminant, the bases of the two affine-power terms are complex conjugates. Their imaginary parts cancel for real x, so the result is purely real with no branch correction needed:

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