User:Dom walden/Multivariate Analytic Combinatorics/Basics

Complex analysis in several variables

Vectors in multidimensional space

The spaces $\mathbb {C} ^{d}$ and $\mathbb {R} ^{d}$ are made up of ordered sets of $n$ complex or real (resp.) numbers

${\textbf {z}}=(z_{1},\cdots ,z_{d})\quad (z_{i}\in \mathbb {C} )$

${\textbf {x}}=(x_{1},\cdots ,x_{d})\quad (x_{i}\in \mathbb {R} )$

We write ${\textbf {z}}\in \mathbb {C} ^{d}$ or ${\textbf {x}}\in \mathbb {R} ^{d}$.

Polydisk and polytorus

For the vectors ${\textbf {a}}\in \mathbb {C} ^{d}$ and ${\textbf {r}}\in \mathbb {R} _{>0}^{d}$ the open polydisk centred at ${\textbf {a}}$ of radius ${\textbf {r}}$

$D_{\textbf {a}}({\textbf {r}})=\{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|<r_{1},\cdots ,|z_{d}-a_{d}|<r_{d}\}$

and the polytorus

$T_{\textbf {a}}({\textbf {r}})=\{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|=r_{1},\cdots ,|z_{d}-a_{d}|=r_{d}\}$

Multivariate Cauchy coefficient formula

This is the multivariate version of the Cauchy coefficient formula.

By the multiple Cauchy integral

$f(z)={\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta _{1}\cdots d\zeta _{n}}{(\zeta _{1}-z)\cdots (\zeta _{n}-z)}}={\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta }{(\zeta -z)}}$

and the fact that

${\frac {1}{\zeta -z}}=\sum _{|k|=0}^{\infty }{\frac {(z-a)^{k}}{(\zeta -a)^{k+1}}}$

we can re-write $f(z)$

$f(z)=\sum _{|k|=0}^{\infty }{\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta }{(\zeta -a)^{k+1}}}(z-a)^{k}=\sum _{|k|=0}^{\infty }c_{k}(z-a)^{k}$

where:

$c_{k}={\frac {1}{(2\pi i)^{n}}}\int _{T}{\frac {f(\zeta )d\zeta }{(\zeta -a)^{k+1}}}$

Domain of convergence

The domain of convergence of a power series is the set of points ${\textbf {z}}\in \mathbb {C} ^{d}$ such that the power series converges absolutely for some neighbourhood of ${\textbf {z}}$.

The associated or conjugate radii of convergence are the vectors ${\textbf {r}}\in \mathbb {R} _{>0}^{d}$ such that the power series converges in the domain $\{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|<r_{1},\cdots ,|z_{d}-a_{d}|<r_{d}\}$ and diverges in the domain $\{{\textbf {z}}\in \mathbb {C} ^{d}:|z_{1}-a_{1}|>r_{1},\cdots ,|z_{d}-a_{d}|>r_{d}\}$. Note that ${\textbf {r}}$ is not necessarily unique and there may be infinite such ${\textbf {r}}$.

In our example, ${\frac {1}{1-x-y}}$, the denominator is zero for $0\leq x\leq 1$ and $y=1-x$.

Multivariate generating functions

Multivariate formal power series

For a generating function in $d$ variables.

${\textbf {z}}=(z_{1},\cdots ,z_{d})\in \mathbb {C} ^{d}$, ${\textbf {n}}=(n_{1},\cdots ,n_{d})\in \mathbb {N} ^{d}$ and ${\textbf {z}}^{\textbf {n}}=z_{1}^{n_{1}}\cdots z_{d}^{n_{d}}$

The multivariate formal power series

$F({\textbf {z}})=\sum _{{\textbf {n}}\in \mathbb {N} ^{d}}f_{\textbf {n}}{\textbf {z}}^{\textbf {n}}=\sum _{(n_{1},\cdots ,n_{d})\in \mathbb {N} ^{d}}f_{n_{1},\cdots ,n_{d}}z_{1}^{n_{1}}\cdots z_{d}^{n_{d}}$

For example

${\frac {1}{1-x-y}}=\sum _{(n,m)\in \mathbb {N} ^{2}}{\binom {n+m}{m}}x^{m}y^{n}=\sum _{n\geq 0}\sum _{m\geq 0}{\binom {n+m}{m}}x^{m}y^{n}$

Diagonals

The central diagonal of a power series

$\Delta F({\textbf {z}})=\sum _{n\geq 0}f_{n,\cdots ,n}z^{n}$

For our example power series, ${\frac {1}{1-x-y}}$, we can represent it in two dimensions like below. The central diagonal is highlighted in green.

	$x^{0}$	$x^{1}$	$x^{2}$	$x^{3}$	$x^{4}$
$y^{0}$	$1x^{0}y^{0}$	$1x^{1}y^{0}$	$1x^{2}y^{0}$	$1x^{3}y^{0}$	$1x^{4}y^{0}$
$y^{1}$	$1x^{0}y^{1}$	$2x^{1}y^{1}$	$3x^{2}y^{1}$	$4x^{3}y^{1}$	$5x^{4}y^{1}$
$y^{2}$	$1x^{0}y^{2}$	$3x^{1}y^{2}$	$6x^{2}y^{2}$	$10x^{3}y^{2}$	$15x^{4}y^{2}$
$y^{3}$	$1x^{0}y^{3}$	$4x^{1}y^{3}$	$10x^{2}y^{3}$	$20x^{3}y^{3}$	$35x^{4}y^{3}$
$y^{4}$	$1x^{0}y^{4}$	$5x^{1}y^{4}$	$15x^{2}y^{4}$	$35x^{3}y^{4}$	$70x^{4}y^{4}$

We can generalise this to a diagonal along a ray r, where ${\textbf {r}}=(r_{1},\cdots ,r_{d})\in \mathbb {N} ^{d}$

$\Delta ^{\textbf {r}}F({\textbf {z}})=\sum _{n\geq 0}f_{nr_{1},\cdots ,nr_{d}}z^{n}$

For example

$\Delta ^{(2,1)}{\frac {1}{1-x-y}}=\sum _{n\geq 0}{\binom {2n+n}{n}}z^{n}$

which is represented below in green.

	$x^{0}$	$x^{1}$	$x^{2}$	$x^{3}$	$x^{4}$
$y^{0}$	$1x^{0}y^{0}$	$1x^{1}y^{0}$	$1x^{2}y^{0}$	$1x^{3}y^{0}$	$1x^{4}y^{0}$
$y^{1}$	$1x^{0}y^{1}$	$2x^{1}y^{1}$	$3x^{2}y^{1}$	$4x^{3}y^{1}$	$5x^{4}y^{1}$
$y^{2}$	$1x^{0}y^{2}$	$3x^{1}y^{2}$	$6x^{2}y^{2}$	$10x^{3}y^{2}$	$15x^{4}y^{2}$
$y^{3}$	$1x^{0}y^{3}$	$4x^{1}y^{3}$	$10x^{2}y^{3}$	$20x^{3}y^{3}$	$35x^{4}y^{3}$
$y^{4}$	$1x^{0}y^{4}$	$5x^{1}y^{4}$	$15x^{2}y^{4}$	$35x^{3}y^{4}$	$70x^{4}y^{4}$

References

• Fuks, B. A. (1963). Theory of Analytic Functions of Several Complex Variables. American Mathematical Society, Providence, Rhode Island.
• Melczer, Stephen (2021). An Invitation to Analytic Combinatorics: From One to Several Variables (PDF). Springer Texts & Monographs in Symbolic Computation.
• Mishna, Marni (2020). Analytic Combinatorics: A Multidimensional Approach. Taylor & Francis Group, LLC.
• Shabat, B. V. (1992). Introduction to Complex Analysis. Part II: Functions of Several Variables. American Mathematical Society, Providence, Rhode Island.