# indices differing by an integer

(0.002 seconds)

## 11—13 of 13 matching pages

##### 11: 21.7 Riemann Surfaces
On this surface, we choose $2g$ cycles (that is, closed oriented curves, each with at most a finite number of singular points) $a_{j}$, $b_{j}$, $j=1,2,\dots,g$, such that their intersection indices satisfy … Next, define an isomorphism $\boldsymbol{{\eta}}$ which maps every subset $T$ of $B$ with an even number of elements to a $2g$-dimensional vector $\boldsymbol{{\eta}}(T)$ with elements either $0$ or $\tfrac{1}{2}$. …
21.7.14 $\boldsymbol{{\eta}}(T_{1}\ominus T_{2})=\boldsymbol{{\eta}}(T_{1})+\boldsymbol% {{\eta}}(T_{2}),$
21.7.15 $4\boldsymbol{{\eta}}^{1}(T)\cdot\boldsymbol{{\eta}}^{2}(T)=\tfrac{1}{2}\left(|% T\ominus U|-g-1\right)\pmod{2},$
##### 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 $\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=\det\begin{bmatrix}w_{1}(z)&w_{2}(% z)\\ w_{1}^{\prime}(z)&w_{2}^{\prime}(z)\end{bmatrix}=w_{1}(z)w_{2}^{\prime}(z)-w_{% 2}(z)w_{1}^{\prime}(z)$

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 $\mathscr{W}\left\{w_{1}(z),\ldots,w_{n}(z)\right\}=\det\left[w_{k}^{(j-1)}(z)\right]$, where $1\leq j,k\leq n$. 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 $a\neq 0,-1,-2,\dots,$ was removed. A clarification regarding the correct constraints for Lerch’s transcendent $\Phi\left(z,s,a\right)$ has been added in the text immediately below. In particular, it is now stated that if $s$ is not an integer then $\left|\operatorname{ph}a\right|<\pi$; if $s$ is a positive integer then $a\neq 0,-1,-2,\dots$; 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 $\zeta\left(s,a\right)=\zeta\left(s,a+m\right)+\sum_{n=0}^{m-1}\frac{1}{(n+a)^{% s}},$ $m=1,2,3,\dots$.
where $h,k$ are integers with $1\leq h\leq k$ and $n=1,2,3,\dots$. …
25.11.33 $h(n)=\sum_{k=1}^{n}k^{-1}.$
25.11.35 $\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=\frac{1}{\Gamma\left(s\right)}% \int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\mathrm{d}x=2^{-s}\left(\zeta% \left(s,\tfrac{1}{2}a\right)-\zeta\left(s,\tfrac{1}{2}(1+a)\right)\right),$ $\Re a>0$, $\Re s>0$; or $\Re a=0$, $\Im a\neq 0$, $0<\Re s<1$.
For an exponentially-improved form of (25.11.43) see Paris (2005b).