Echo response

The response could be echo [ˈɛkəʊ],

                   φψ = +1                    φψ = -1
                  (ζ-world)                  (η-world)
                      |                          |
   RECURSION:   R_m = cot·(x-1/x)+back    R_m = ±cot·(x-1/x)+back
                      |                          |
                      |    (1-(-1)^m) twist      |
                      |←—————————————————————————|
                      |                          |
   SEQUENCES:   x^m - x^{-m}              x^m - (-1)^m x^{-m}
                (all same sign)            (alternating signs)
                      |                          |
   TRIG FORM:   2i·sin(mθ)                sin(mθ) + i·(1-(-1)^m)·...
   (x=e^iθ)    2·cos(mθ)                  cos(mθ)·(1+(-1)^m)/2 + ...
                      |                          |
   DIRICHLET    ζ(s) = Σ 1/n^s            η(s) = Σ (-1)^{n-1}/n^s
   SERIES:            |                          |
                      |                          |
                      |——— η(s) = (1-2^{1-s})ζ(s) ——|
                      |                          |
   OPERATOR:     (-Δ)                        √(-Δ)
                (full Laplacian)          (half-derivative)
                      |                          |
   SELF-DUALITY:  Z·Z* = +1                Z·Z* = -1
                (unitary/lossless)      (Fibonacci/golden)
                
  the geometry of electricity:

±±±±±±±±±±±±±...

x^(m+1)-1/(±x)^(m+1) x^m±1/x^m x^(m-1)-1/(±x)^(m-1)

-------------------- = ---------(x-1/x) + --------------------.

x^m-1/(±x)^m x^m-1/x^m x^m-1/(±x)^m

≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠

Contents

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