A Set of 4 { Quad-Symmetric Based } Functions

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

Top Of Page

Section I - The Search for { Quad-Symmetric Based } Functions

Summary for the Section :

• In this section , we shall investigate a set of 4 related { quad-symmetric based } functions ,
  • which wound then enable us to express the { Sine Function } , the { Cosine Function } and the { Exponential Function } ,
    • as a { linear sum } of the 4 said-functions .

go to Top Of Page

Setting up the 4 { Quad-Symmetric Based } functions :

• Let us now look at a set of { 4 functions } for the variable [ z ] , namely the functions :
  • 

    such that the 4 functions satisfy this { defining property } below , namely :

    • 

      

      

      

  And this set of { 4 functions } is then known as { quad-symmetric based } functions simply because :

  • the { derivative relations } here are always circular in nature , so that :
    • it does not matter which of the 4 functions is chosen as the { lead-off function } for analysis ,

      and the result will always be the same .

    And it is to this extent that the { 4 functions } are { quad-symmetric } .

go to Top Of Page

Imposing a Special Condition :

• For our first analysis here , let us impose a { Special Condition } on the set of { 4 functions } , namley the condition that :
  • 

    

    i.e. :

    • [ Psi-A ] and [ Psi-C ] are { additive conjugates } of one-another , and

      [ Psi-B ] and [ Psi-D ] are { additive conjugates } of one-another .

• We shall now assume , for our analysis purposes , that there exists a particular value of [ z ] , such that :
  • the value of [ Psi-A ] and [ Psi-B ] are the same at this value [ Z-zero ] .

  We can then write :

  • 

    where [ K ] is the value of [ Psi-A ] and [ Psi-B ] at { [ z ] = [ Z-zero ] } .

  Consequently :

  • 

    arising from the { additive conjugate } relations we have above .

• For the trivial case of [ K = 0 ] , we simple have the case :
  • 

    and all { derivatives } are therefore also [ zero ] .

    No problems here as the circular { derivative relations } are always satisfied .

• For the case of [ K ] not being equal to [ zero ] , we have :
  • 

    

    

    

    and so-on-and-so-forth , with the pattern [ +K , -K , -K, + K ] repeating itself ad infinitum .

  Let us now do a { Taylor Series } expansion here , via :

  • 

    to yield :

  • 

    

    

    

  And the consolidated format of these 4 equations is then :

  • 

    

    

    

  And we simply note here that above format is the same format for the { Taylor Series } expansion for :

  • the { sine / cosine / minus-sine / minus-cosine } functions at the point { [ Z-zero ] = [ pi / 4 ] } ,
    • as follows :

  • 

    

    

    

    But more on this later in the next Section .

go to Top Of Page

The Exponential Function :

• Let us now remove the { Special Condition } that the 4 functions be 2 pairs of { additive conjugates } ,
  • and we notice immediately that the { exponential function } does satisfy the original { Defining Property } above .

• We then set up :
  • 

    and consequently :

  • 

    

    

    

• Let us now do a { Taylor Series } expansion from the point { [ Z-zero ] = [ 0 ] } , i.e. :
  • 

  We then have :

  • 

    or ,

  • 

go to Top Of Page

Setting up the { Q-Functions } :

• We notice , at this point , that the { infinite series } above can be split into four (4) { infinite series's } ,
  • and we now set up the 4 new { Q-Functions } as follows :
    • 

      

      

      

    and we have :

    • 

      

      

      

      with these satisfying the original { defining property } for the { Quad-Symmetric Based } functions .

• We can then write :
  • 

    and ,

  • 

    

    

    

go to Top Of Page

• Rather convenient , but more to come , next .

go to the next section : Section II - The 4 { Q-Functions }

return to { Quad-Symmetric Based Functions } HomePage

original dated 2008-3-23