A Set of 4 { Quad-Symmetric Based } Functions

by Frank C. Fung - 1st published in March, 2008.

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Section II - The 4 { Q-Functions }

Summary for the Section :

• 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 } .

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Graphics for the 4 { Q-Functions } :

• 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 ) .

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Sine and Cosine for Complex Variables :

• 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 :
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