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11: 21.7 Riemann Surfaces
On this surface, we choose 2 g cycles (that is, closed oriented curves, each with at most a finite number of singular points) a j , b j , j = 1 , 2 , , g , such that their intersection indices satisfy … Next, define an isomorphism η which maps every subset T of B with an even number of elements to a 2 g -dimensional vector η ( T ) with elements either 0 or 1 2 . …
21.7.12 T 1 T 2 = ( T 1 T 2 ) ( T 1 T 2 ) .
21.7.14 η ( T 1 T 2 ) = η ( T 1 ) + η ( T 2 ) ,
21.7.15 4 η 1 ( T ) η 2 ( T ) = 1 2 ( | T U | - g - 1 ) ( mod 2 ) ,
12: Errata
  • Section 14.6(ii)

    A sentence was added at the end of the subsection indicating that for generalizations, see Cohl and Costas-Santos (2020).

  • Section 1.13

    In Equation (1.13.4), the determinant form of the two-argument Wronskian

    1.13.4 𝒲 { w 1 ( z ) , w 2 ( z ) } = det [ w 1 ( z ) w 2 ( z ) w 1 ( z ) w 2 ( z ) ] = w 1 ( z ) w 2 ( z ) - w 2 ( z ) w 1 ( z )

    was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the n -argument Wronskian is given by 𝒲 { w 1 ( z ) , , w n ( z ) } = det [ w k ( j - 1 ) ( z ) ] , where 1 j , k n . Immediately below Equation (1.13.4), a sentence was added giving the definition of the n -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for n th-order differential equations. A reference to Ince (1926, §5.2) was added.

  • Equation (25.14.1)

    the previous constraint a 0 , - 1 , - 2 , , was removed. A clarification regarding the correct constraints for Lerch’s transcendent Φ ( z , s , a ) has been added in the text immediately below. In particular, it is now stated that if s is not an integer then | ph a | < π ; if s is a positive integer then a 0 , - 1 , - 2 , ; if s is a non-positive integer then a can be any complex number.

  • Section 10.8

    A sentence was added just below (10.8.3) indicating that it is a rewriting of (16.12.1).

    Suggested by Tom Koornwinder.

  • References

    An addition was made to the Software Index to reflect a multiple precision (MP) package written in C++ which uses a variety of different MP interfaces. See Kormanyos (2011).

  • 13: 25.11 Hurwitz Zeta Function
    25.11.4 ζ ( s , a ) = ζ ( s , a + m ) + n = 0 m - 1 1 ( n + a ) s , m = 1 , 2 , 3 , .
    where h , k are integers with 1 h k and n = 1 , 2 , 3 , . …
    25.11.33 h ( n ) = k = 1 n k - 1 .
    25.11.35 n = 0 ( - 1 ) n ( n + a ) s = 1 Γ ( s ) 0 x s - 1 e - a x 1 + e - x d x = 2 - s ( ζ ( s , 1 2 a ) - ζ ( s , 1 2 ( 1 + a ) ) ) , a > 0 , s > 0 ; or a = 0 , a 0 , 0 < s < 1 .
    For an exponentially-improved form of (25.11.43) see Paris (2005b).