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Consider the simplest case: polynomial substitution, with polynomials that are not units (that is, their constant coefficient is zero).
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We can obtain the same result by substituting the actual polynomials in, rather than the polynomials converted to power series.
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We can verify that the result is correct by comparing with the result of doing the substitution entirely in the domain of polynomials.
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We can do the same with unit power series.
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The following example shows that simple arithmetic does not suffice to do the substitution operation: the input power series has rational coefficients, but the result has transcendental coefficients. We compute the power series for from that of .
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To underscore the necessity to have analytic expressions in the presence of unit power series, consider the following example, where we omit the analytic expression for the series of .
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We want to substitute for and for in . If we first substitute for , then the resulting power series doesn't know its analytic expression, and we obtain an error when we substitute for .
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In the other order, the process does work.
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This is the order chosen by Maple when the two substitutions are given simultaneously.
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