Mathematics - Properties of Summation Symbols

1. single sum

$$\sum _{i=1}^{n}f(x_i)=f(x_1)+f(x_2)+\cdots +f(x_n)$$

1.1 性质1,提取公因式

若$h(y,z)$的取值和x无关,则有：
$$\sum _{i=1}^{n}h(y,z)f(x_i)=h(y,z)\sum _{i=1}^{n}f(x_i)$$
将变量$i$写成$x_i$更形象：
$$\sum _{x_i}h(y,z)f(x_i)=h(y,z)\sum _{x_i}f(x_i)$$
Abbreviated above,实际上$x_i\in X$,$X=\{x_1,x_2,\cdots ,x_n\}$:
$$\sum _{x_i\in X}Usually\; can\; be\; abbreviated\; as\sum _{x_i},Indicates\; to\; accumulate\; all{x}_i可取的值$$

2. 多重求和

Take the double summation as an example：
$$\sum _{i=1}^n\sum _{j=1}^{m}f(x_i)h(y_j)=f(x_1)\sum _{j=1}^{m}h(y_j)+f(x_2)\sum _{j=1}^{m}h(y_j)+\cdots +f(x_n)\sum _{j=1}^{m}h(y_j)=If\; it\; is\; expanded\; again,\; it\; will\; be\; omitted$$

2.1 性质1,The symbol order can be changed

两重：

$$\sum _{i=1}^n\sum _{j=1}^{m}f(x_i)h(y_j)=\sum _{j=1}^m\sum _{i=1}^{n}f(x_i)h(y_j)$$
注意,When the range of a summation is limited by another variable,The commutative law does not apply,如：
$$\sum _{i=1}^n\sum _{j=1}^{i}f(x_i)h(y_j)\ne \sum _{j=1}^i\sum _{i=1}^{n}f(x_i)h(y_j)$$
多重：
$$\sum _{x_i}\sum _{y_j}\sum _{z_k}{f}_1(x_i)f_2(y_j)f_3(z_k)=\sum _{z_k}\sum _{y_j}\sum _{x_i}{f}_1(x_i)f_2(y_j)f_3(z_k)$$
$f_1(x_i)f_2(y_j)f_3(z_k)$Can be seen as a function$f(x_1,x_2,x_3)$,A more general form is obtained：

$$\sum _{x_i}\sum _{y_j}\sum _{z_k}f(x_1,x_2,x_3)=\sum _{z_k}\sum _{y_j}\sum _{x_i}f(x_1,x_2,x_3)$$
Keep the premise of interchangeability in mind：x,y,z的取值范围,相互没有影响.

2.1 性质2,Symbols can be found separately

有时候,为了求解的方便,We don't want functions$f(x_1,x_2,x_3)$written as a whole,Instead, they are separated and evaluated separately.
$$\sum _{x_i}\sum _{y_j}\sum _{z_k}{f}_1(x_i)f_2(y_j)f_3(z_k)=\sum _{x_i}{f}_1(x_i)\sum _{y_j}{f}_2(y_j)\sum _{z_k}{f}_3(z_k)$$
x,y,zThe range of values ​​should also satisfy each other without affecting each other,A simple proof can be done by expanding the calculation.
The advantage of the above properties is that,Complex problems can be divided into three parts and calculated separately,再求乘积.

$$(\sum _{x_i}{f}_1(x_i))(\sum _{y_j}{f}_2(y_j))(\sum _{z_k}{f}_3(z_k))$$