A Set of 4 { Quad-Symmetric Based } Functions

Section II - The 4 { Q-Functions }

In this section , we shall look at the graphs of the 4 { Q-Functions } ,
- and develop some rather useful formulas for { sine } and { cosine } on the { complex plane } .

Let us now look at graphs for the 4 { Q-Functions } , in the order of the { derivatives } ,
- i.e. the order { [ Q0 ] - [ Q3 ] - [ Q2 ] - [ Q1 ] } .
Let us now bring { [ Q0 ] and [ Q2 ] } together and { [ Q3 ] and [ Q1 ] } together ,
- And we notice here that each pair of { lines } are indeed quite close together .
Let us now enlarge these a little aroud the [ origin ] for a better view :
- And in general , the [ absolute value ] of the difference between each pair of { lines } is always { less than or equal to [ 1 ] }
  
  ( see below for an explanation ) .

Let us now use these 4 { Q-Functions } for our expansion of the { Sine } and { Cosine } functions ,
- and we start-off with [ Z-zero ] being an element of :
For the { Sine } function , we have :
For the { Cosine } function , we have :
Thus , in general , the expansion formulas for a { real variable } [ sigma ] can be written as follows :
- or ,
We take note at this point the following relations :
Thus , if the value of [ x ] is purely { imaginary } , while [ sigma ] remains { real } , then the functions :
- { sine of [ sigma + x ] } and { cosine of [ sigma + x ] } would :
  - involve only the term :
    - for the { real part } of the final value , and
  - involve only the term :
    - for the { imaginary part } of the final value .
  And to the extent that the [ absolute value ] of either the { sine } or { cosine } function cannot exceed [ 1 ] ,
  - the value of either of these terms , also , may not exceed [ 1 ] .