of one variable
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11: 19.17 Graphics
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►Because the -function is homogeneous, there is no loss of generality in giving one variable the value or (as in Figure 19.3.2).
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12: 1.10 Functions of a Complex Variable
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§1.10(i) Taylor’s Theorem for Complex Variables
… ►Analytic continuation is a powerful aid in establishing transformations or functional equations for complex variables, because it enables the problem to be reduced to: (a) deriving the transformation (or functional equation) with real variables; followed by (b) finding the domain on which the transformed function is analytic. …13: 19.16 Definitions
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►When one variable is 0 without destroying convergence, any one of (19.16.14)–(19.16.17) is said to be complete and can be written as an -function with one less variable:
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14: Bibliography F
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The third Appell function for one large variable.
J. Approx. Theory 165, pp. 60–69.
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The second Appell function for one large variable.
Mediterr. J. Math. 10 (4), pp. 1853–1865.
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15: 10.73 Physical Applications
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►and on separation of variables we obtain solutions of the form , from which a solution satisfying prescribed boundary conditions may be constructed.
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►On separation of variables into cylindrical coordinates, the Bessel functions , and modified Bessel functions and , all appear.
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16: Annie A. M. Cuyt
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►A lot of her research has been devoted to rational approximations, in one as well as in many variables, and sparse interpolation.
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17: Mathematical Introduction
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►Special functions with one real variable are depicted graphically with conventional two-dimensional (2D) line graphs.
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18: 36.12 Uniform Approximation of Integrals
19: 9.14 Incomplete Airy Functions
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►Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter.
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20: 31.2 Differential Equations
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►where and with are generators of the lattice for .
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►By composing these three steps, there result possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1).
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►If is one of the homographies that map to , then satisfies (31.2.1) if is a solution of (31.2.1) with replaced by and appropriately transformed parameters.
…If is one of the homographies that do not map to , then an appropriate prefactor must be included on the right-hand side.
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►Each is a substitution of dependent and/or independent variables that preserves the form of (31.2.1).
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