Partial fraction theorem

Applying the general partial fraction theorem, Eq. (3-72), to Eq. (3-96) and then taking inverse transforms gives Eq. (3-99). 

Benet, L.Z. Turi, J.S. Use of general partial fraction theorem for obtaining inverse laplace transforms in pharmacokinetic analysis. J. Pharm. Sci. 1971, 60, 1593-1594. 

B. PARTIAL-FRACTIONS EXPANSION. The linearity theorem [Eq. (18.36)] permits us to expand the function into a sum of simple terms and invert each individually. This is completely analogous to Laplace-transformation inversion. Let F, be a ratio of polynomials in z, Mth-order in the numerator and iVth-order in the denominator. We factor the denominator into its N roots pi, P2, Ps,... 

Forcing function, 143 periodic, 144 transient, 143 Fourier transform, 170 Fractional time, 29 Fractionation factor, 301 Fraction theorem, general partial, 85 Frame, rotating, 170 Franck-Condon principle, 435 Free energy, 211 transfer, 418... 

The Together statement collects all terms of an expression together over a common denominator, while the Apart statement breaks the expression apart into terms with simple denominators, as in the method of partial fractions. The theorem of partial fractions states that if Q(x) can be factored in the form... 

The fundamental formula of the method of partial fractions is a theorem of algebra that says that if Q(x) is given by Eq. (5.45) and P x) is of lower degree than Q x), then... 

We have derived the general Inversion theorem for pole singularities using Cauchy s Residue theory. This provides the fundamental basis (with a few exceptions, such as /s) for inverting Laplace transforms. However, the useful building blocks, along with a few practical observations, allow many functions to be inverted without undertaking the formality of the Residue theory. We shall discuss these practical, intuitive methods in the sections to follow. Two widely used practical approaches are (1) partial fractions, and (2) convolution. 

The previous example is very straightforward and could easily be inverted by use of partial fractions and a table of Laplace transforms. This next example may be tabulated. However, it is used here to demonstrate more clearly how the residue theorem may be useful for similar or more complicated inversions. Consider... 

Using the partial fractions (or Heaviside theorem), we can obtain the inverse Laplace transform. 

We present here an extremely useful theorem for modeling of heterogenous reactions. This theorem is used in particular to know if variables such as temperature, partial pressures of gases, and fractional extent are separate in the expression of the rate, that is, if, for example, the latter can be or cannot be put in the form of a product of functions with only one variable of each one, then we have ... 

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