By W. J. Thron

ISBN-10: 0387167684

ISBN-13: 9780387167688

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**Additional resources for Analytic Theory of Continued Fractions II**

**Sample text**

U;n)) of the Fourier transform of {$(x; n)). The inverse Fourier transform can be defined in exactly the same way since {9(u;n)) is also a regular sequence. - A result that can also be derived from Parseval's Theorem and from the symmetry of the Fourier transform as follows. P are Fourier transform pairs and if t$ E S(R,C ) and q5 and Q1 are Fourier transform pairs, then From this theorem, using the algebra of generalised functions, it can be shown that the usual properties of the classical Fourier transform are preserved.

By a similar means it can be shown that Analytic Methods for Partial Differential Equations 32 and more generally where again C is traversed in the anticlockwise sense. A consequence of these results is Taylor's theorem which states that if f ( z ) is an analytic function regular in a neighbourhood of z = a, then f ( z ) may be expanded in the form 00 - - f(z) = 1a n ( . - a)". t. lz - a1 < R for some R > 0. 20 to yield n f ( a + h ) = f ( a )+ r=l hr f (')(a) r! + remainder. 22) If a function is not regular in the domain 1% - a1 < Rl but is regular in the annulus Rz < lz - a1 < R1 then the expansion has the form f ( r )= a,(% - a)" + bn(z - a)-" which is Laurent's theorem.

40 Analytic Methods for Partial Differential Equations For the delta function, the set of test functions can be extended to consider bounded and piecewise continuous functions. Since the Heaviside step function defines a generalised function, then for 4 E S(R,C) it follows that This property of the delta function is called the sampling property and is arguably its most important property which is why some authors define the 6 function via its sampling property alone. In other words, we define the delta function in terms of the role it plays in a mathematical operation rather than in terms of what it actually is.