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1: 1.9 Calculus of a Complex Variable
Continuity
That is, given any positive number ϵ , however small, we can find a positive number δ such that | f ( z ) f ( z 0 ) | < ϵ for all z in the open disk | z z 0 | < δ . A function of two complex variables f ( z , w ) is continuous at ( z 0 , w 0 ) if lim ( z , w ) ( z 0 , w 0 ) f ( z , w ) = f ( z 0 , w 0 ) ; compare (1.5.1) and (1.5.2). …
2: 16.13 Appell Functions
The following four functions of two real or complex variables x and y cannot be expressed as a product of two F 1 2 functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): …
3: 19.25 Relations to Other Functions
19.25.11 E ( ϕ , k ) = 1 3 k 2 R D ( c k 2 , c , c 1 ) + c k 2 / ( c c 1 ) , ϕ 1 2 π .
4: Mathematical Introduction
Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. …
5: 31.1 Special Notation
x , y real variables.
z , ζ , w , W complex variables.
a complex parameter, | a | 1 , a 1 .
q , α , β , γ , δ , ϵ , ν complex parameters.
Sometimes the parameters are suppressed.
6: 28.15 Expansions for Small q
28.15.3 me ν ( z , q ) = e i ν z q 4 ( 1 ν + 1 e i ( ν + 2 ) z 1 ν 1 e i ( ν 2 ) z ) + q 2 32 ( 1 ( ν + 1 ) ( ν + 2 ) e i ( ν + 4 ) z + 1 ( ν 1 ) ( ν 2 ) e i ( ν 4 ) z 2 ( ν 2 + 1 ) ( ν 2 1 ) 2 e i ν z ) + ;
7: 8.11 Asymptotic Approximations and Expansions
§8.11 Asymptotic Approximations and Expansions
For bounds on R n ( a , z ) when a is real and z is complex see Olver (1997b, pp. 109–112). … See Tricomi (1950b) for these approximations, together with higher terms and extensions to complex variables. … For (8.11.18) and extensions to complex values of x see Buckholtz (1963). …
8: 19.21 Connection Formulas
19.21.1 R F ( 0 , z + 1 , z ) R D ( 0 , z + 1 , 1 ) + R D ( 0 , z + 1 , z ) R F ( 0 , z + 1 , 1 ) = 3 π / ( 2 z ) , z ( , 0 ] .
9: 4.2 Definitions
ln z is a single-valued analytic function on ( , 0 ] and real-valued when z ranges over the positive real numbers. … In the DLMF we allow a further extension by regarding the cut as representing two sets of points, one set corresponding to the “upper side” and denoted by z = x + i 0 , the other set corresponding to the “lower side” and denoted by z = x i 0 . …Consequently ln z is two-valued on the cut, and discontinuous across the cut. … The function exp is an entire function of z , with no real or complex zeros. … This is an analytic function of z on ( , 0 ] , and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless a . …
10: Bibliography B
  • E. Barouch, B. M. McCoy, and T. T. Wu (1973) Zero-field susceptibility of the two-dimensional Ising model near T c . Phys. Rev. Lett. 31, pp. 1409–1411.
  • H. A. Bethe and E. E. Salpeter (1957) Quantum mechanics of one- and two-electron atoms. Springer-Verlag, Berlin.
  • H. A. Bethe and E. E. Salpeter (1977) Quantum Mechanics of One- and Two-electron Atoms. Rosetta edition, Plenum Publishing Corp., New York.
  • P. Boalch (2006) The fifty-two icosahedral solutions to Painlevé VI. J. Reine Angew. Math. 596, pp. 183–214.
  • S. Bochner and W. T. Martin (1948) Several Complex Variables. Princeton Mathematical Series, Vol. 10, Princeton University Press, Princeton, N.J..