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Topic: Nbic Functions and their tan-Formulas
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Doctor Nisith Bairagi

Posts: 23
From: Uttarpara, West Bengal, India
Registered: 3/2/13
Nbic Functions and their tan-Formulas
Posted: Apr 4, 2013 2:22 AM
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From: Doctor Nisith Bairagi
Subject: Nbic Functions and their tan- Formulas
Date: April 4, 2013
(sci.math.independent)
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Dear Readers,
Presented here are few typical varieties of Nbic functions, and indicated how they could be generated. Though these Nbic functions, (Single and their higher varieties like, Double, Triple, etc.), could be generated in a number of alternative ways, only few simpler representative ones are demonstrated here. Striking similarities exit between the Nbic function formulas, and the corresponding trigonometric ones. Only tan-formulas here in the form, tan(A + B) = (tan A + tan B)/[1-tan A.tan B], is focused here.
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Nbic function::
Form-wise, they appear as the sum of the basic unit functions, each of which is a product of trigonometric and hyperbolic functions, [like, sin(x).cosh(y), cos (x).sinh(-y), etc.]. They may be of several varieties, starting from, Single Nbic Function (SNF), and onwards to the higher ones, as Double Nbic Function (DNF), Triple Nbic Function (TNF), etc.
The pattern of the Nbic functions during generation, show striking similarity with the type of trigonometric expansion of sin(A + B) = (sinA.cosB + cosA.sinB), cos(A + B) = (cosA.cosB-sinA.sinB), and their pattern of tan-formulas, tan(A + B) = (tan A + tan B)/[1-tan A.tan B].

Typical Circular functions are: [sin(x), cos(x)], Hyperbolic functions are: [sinh(x), cosh(x)]. Similarly, keeping tradition, Nbic functions are: [sinN(x, y), cosN(x, y)], etc.
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Single Nbic Function (SNF), N<1>(x, +/-y)::
They are generated by separating the Real and Imaginary parts from the product of: Coci (Complex circular) function, e^(+/-)(ix) = (cos x +/- i sin x), and, Cohy (Complex hyperbolic) function, f^(+/-)(iy) = (cosh y +/- i sinh y).

Thus: e^(+/-)(ix).f^(+/-)(iy) = cosN(x, +/-y) + i sinN(x, +/-y), yields:
cosN(x, +/-y) = cos x cosh y -/+ sin x sinh y
sinN(x, +/-y) = sin x cosh y +/- cos x sinh y
tanN(x, +/-y) = sinN(x, +/-y)/cosN(x, +/-y) = (tan x +/-tanh y)/(1-/+ tan x.tanh y)
from which, putting y = x, get:: sinN(x, +/-x), and, cosN(x, +/-x), and so,
tanN(x, +/-x) = (tan x +/-tanh y)/(1-/+ tan x.tanh x).
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Double Nbic Function (DNF), N<2>(x, +/-y)::
They are generated by separating the Real and Imaginary parts from the product of: SNF and Coci, or, Cohy function.
[Suffixes:: (2) for Double, (e) or (f) indicate post-multiplier function is (Coci) or (Cohy), (1) or (2) indicate pre-multiplier Nbic function of (x, y) or (x, x), respectively.]

Their typical varieties are:
[cosN<2e1>(x, +/-y), and, sinN<2e1>(x, +/-y)], are obtained from the product of [cosN(x, +/-y) +/- i sinN(x, +/-y)], when post multiplied by, e^(+/-)(iy).
For typical example:
[cosN<2e1>(x, y), sinN<2e1>(x, y)]
= [cosN(x. y), sinN(x, y)].cos y -/+ [sinN(x, y), cosN(x, y)].sin y
tanN<2e1>(x, y) = [tan(x + y) + tanh y]/[1- tan(x + y).tanh y]
tanN<2e1>(x, x) = [tan 2x + tanh x]/[1- tan 2x.tanh x]
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[cosN<2e2>(x, +/-y), and, sinN<2e2>(x, +/-y)], are obtained from the product of [cosN(x, +/-x) +/- i sinN(x, +/-x)], when post multiplied by, e^(+/-)(iy).
For typical example:
[cosN<2e2>(x, y), sinN<2e2>(x, y)]
= [cosN(x, x), sinN(x, x)].cos y -/+ [sinN(x, x), cosN(x, x)].sin y
tanN<2e2>(x, y) = [tan(x + y) + tanh x]/[1- tan(x + y).tanh x]
tanN<2e2>(x, x) = [tan 2x + tanh x]/[1- tan 2x.tanh x]
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tanN<2e1>(x, x) = tanN<2e2>(x, x)
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[cosN<2f1>(x, +/-y), and, sinN<2f1>(x, +/-y)], are obtained from the product of
[cosN(x, +/-y) +/- i sinN(x, +/-y)], when post multiplied by, f^(+/-)(iy).
For typical example:
[cosN<2f1>(x, y), sinN<2f1>(x, y)]
= [cosN(x. y), sinN(x, y)].cosh y -/+ [sinN(x, y), cosN(x, y)].sinh y
tanN<2f1>(x, y) = (tan x + sinh 2y)/(1-tan x.sinh 2y)
tanN<2f1>(x, x) =(tan x + sinh 2x)/(1-tan x.sinh 2x)
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[cosN<2f2>(x, +/-y), and, sinN<2f2>(x, +/-x)], are obtained from the product of
[cosN(x, +/-x) +/- i sinN(x, +/-x)], when post multiplied by, f^(+/-)(iy).
For typical example:
[cosN<2f2>(x, y), sinN<2f2>(x, y)]
= [cosN(x, x), sinN(x, x)].cosh y -/+ [sinN(x, x), cosN(x, x)].sinh y
tanN<2f2>(x, y) = [tan x + sinh(x + y)/cosh(x-y)]/[1-tan x.sinh(x + y)/cosh(x-y)]
tanN<2f2>(x, x) =(tan x + sinh 2x)/(1-tan x.sinh 2x)
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tanN<2f1>(x, x) = tanN<2f2>(x, x)
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Triple Nbic Function (TNF), N<3>(x, +/-y)::
There are several alternative ways TNF-s can be generated.
It may be obtained also, by post multiplying by Coci or Cohy functions,
on N<2>(x, +/-y), or, N<2>(x, +/-x), and so on.
It may be also generated as follows:
For typical example:
[cosN<3e1>(x, y), sinN<3e1>(x, y)]
= [cosN(x. x), sinN(x, x)].cosN(y, y) -/+ [sinN(x, x), cosN(x, x)].sinN(y, y)
tan N<3e1>(x, y) = [tanN(x, x) + tanN(y, y)]/[1- tanN(x, x).tanN(y, y)]
and so on.
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For elaborate details, please refer to the Book : <<Advanced Trigonometric Relations through Nbic Functions>> by Nisith K Bairagi, New Age International Publishers, New Delhi, 2012, ISBN:978-81-224-3023-3.
Can any of my readers point out the field of practical application of the proposed Nbic functions? Please communicate/reply.
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Thanks to All.
Doctor Nisith Bairagi
<bairagi605@yahoo.co.in>
Uttarpara, West Bengal, India
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