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Mathematics for Enzyme Reaction Kinetics and Reactor Performance. F. Xavier MalcataЧитать онлайн книгу.

Mathematics for Enzyme Reaction Kinetics and Reactor Performance - F. Xavier Malcata


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in denominator would be increased, and in either case the multiplicity of r1 would not equal s1 (as postulated). Hence, one may arbitrarily define (the still unknown) constant Α1,1 as

      (2.197)equation

      In view of Eq. (2.159), r1 being a root of Y<m supports

      (2.198)equation

      after dropping xr1 from both numerator and denominator of the first term in the right‐hand side; ς<m−1/images is again a regular rational fraction, because the degree of ς<m−1{x} is lower than m − 1 (as indicated by the subscript utilized) – while the degree of the corresponding polynomial in denominator equals s1 – 1 + (ms1) = m − 1, on account of the degree s1 − 1 of images and the degree ms1 of images. Therefore, one may proceed to another splitting step of the type

      with Α1,2 denoting a second constant to be replaced by

      (2.202)equation

      (2.203)equation

      as long as images, following Eqs. (2.199) and (2.200) as template. This method may undergo up to s1 iterations, to eventually produce

      (2.204)equation

      The same rationale may then be applied to the second root r2, of multiplicity s2, and so on, until one gets

      therefore, any proper rational fraction with poles r1, r2, …, rs (or r, for short) of multiplicity s1, s2, …, ss, respectively (or s for short), may be expanded as a sum of partial fractions bearing a constant in numerator, as well as xr, (xr)2, …, (xr)s sequentially in denominator – irrespective of the mathematical nature of such roots.

      To avoid emergence of complex numbers – and taking advantage of the fact that if a polynomial with real coefficients has complex roots then they always appear as conjugate pairs (otherwise its coefficients would necessarily be complex numbers), one may lump pairs of complex partial fractions as

      (2.206)equation

      upon elimination of parentheses in numerator, and rearrangement of inner parentheses in denominator, one gets

      (2.210)equation

      which may be rewritten as

      (2.211)equation

      the new constants are defined as

      (2.212)Скачать книгу

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