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real and imaginary parts

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1: 19.3 Graphics
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Figure 19.3.7: K ( k ) as a function of complex k 2 for - 2 ( k 2 ) 2 , - 2 ( k 2 ) 2 . … Magnify 3D Help
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Figure 19.3.8: E ( k ) as a function of complex k 2 for - 2 ( k 2 ) 2 , - 2 ( k 2 ) 2 . … Magnify 3D Help
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Figure 19.3.9: ( K ( k ) ) as a function of complex k 2 for - 2 ( k 2 ) 2 , - 2 ( k 2 ) 2 . … Magnify 3D Help
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Figure 19.3.10: ( K ( k ) ) as a function of complex k 2 for - 2 ( k 2 ) 2 , - 2 ( k 2 ) 2 . … Magnify 3D Help
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Figure 19.3.12: ( E ( k ) ) as a function of complex k 2 for - 2 ( k 2 ) 2 , - 2 ( k 2 ) 2 . … Magnify 3D Help
2: 29.16 Asymptotic Expansions
The approximations for Lamé polynomials hold uniformly on the rectangle 0 z K , 0 z K , when n k and n k assume large real values. …
3: 7.9 Continued Fractions
7.9.1 π e z 2 erfc z = z z 2 + 1 2 1 + 1 z 2 + 3 2 1 + 2 z 2 + , z > 0 ,
7.9.2 π e z 2 erfc z = 2 z 2 z 2 + 1 - 1 2 2 z 2 + 5 - 3 4 2 z 2 + 9 - , z > 0 ,
7.9.3 w ( z ) = i π 1 z - 1 2 z - 1 z - 3 2 z - 2 z - , z > 0 .
4: 28.25 Asymptotic Expansions for Large z
28.25.4 z + , - π + δ ph h + z 2 π - δ ,
28.25.5 z + , - 2 π + δ ph h + z π - δ ,
5: 19.32 Conformal Map onto a Rectangle
with x 1 , x 2 , x 3 real constants, has differential
19.32.2 d z = - 1 2 ( j = 1 3 ( p - x j ) - 1 / 2 ) d p , p > 0 ; 0 < ph ( p - x j ) < π , j = 1 , 2 , 3 .
z ( x 3 ) = R F ( x 3 - x 1 , x 3 - x 2 , 0 ) = - i R F ( 0 , x 1 - x 3 , x 2 - x 3 ) .
As p proceeds along the entire real axis with the upper half-plane on the right, z describes the rectangle in the clockwise direction; hence z ( x 3 ) is negative imaginary. …
6: 23.1 Special Notation
𝕃

lattice in .

z = x + i y

complex variable, except in §§23.20(ii), 23.21(iii).

7: 27.14 Unrestricted Partitions
The corresponding unrestricted partition function is denoted by p ( n ) , and the summands are called parts; see §26.9(i). … The number of partitions of n into at most k parts is denoted by p k ( n ) ; again see §26.9(i). … where ε = exp ( π i ( ( ( a + d ) / ( 12 c ) ) - s ( d , c ) ) ) and s ( d , c ) is given by (27.14.11). …
27.14.17 Δ ( a τ + b c τ + d ) = ( c τ + d ) 12 Δ ( τ ) ,
The 24th power of η ( τ ) in (27.14.12) with e 2 π i τ = x is an infinite product that generates a power series in x with integer coefficients called Ramanujan’s tau function τ ( n ) : …
8: 23.11 Integral Representations
provided that - 1 < ( z + τ ) < 1 and | z | < τ .
9: 5.4 Special Values and Extrema
5.4.16 ψ ( i y ) = 1 2 y + π 2 coth ( π y ) ,
5.4.18 ψ ( 1 + i y ) = - 1 2 y + π 2 coth ( π y ) .
10: 4.2 Definitions
The real and imaginary parts of ln z are given by …
4.2.29 | z a | = | z | a exp ( - ( a ) ph z ) ,
4.2.30 ph ( z a ) = ( a ) ph z + ( a ) ln | z | ,