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Topic: Complex Circular (Coci) and Complex Hyperbolic (Cohy) Numbers
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Doctor Nisith Bairagi

Posts: 27
From: Uttarpara, West Bengal, India
Registered: 3/2/13
Complex Circular (Coci) and Complex Hyperbolic (Cohy) Numbers
Posted: Mar 30, 2013 9:41 AM
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From: Doctor Nisith Bairagi
Subject: Complex Circular (Coci) and Complex Hyperbolic (Cohy) Numbers
Date: March 30, 2013
Dear Readers,
Please comment on the new proposal for representing the complex numbers. Your constructive criticism on this topic will be an asset.

A complex number [a (+/-) ib], can be represented through several ways, viz.,
Complex Circular (Coci) form, and Complex Hyperbolic form, and accordingly, the number may be identified as Coci number or Cohy number. While the Coci form is well known, the Cohy form, is presented as a new proposal.

Coci number:
Using exponential function, e^c = (cosh c + sinh x), we write:
Coci number: a (+/-) ib = R [cos c (+/-) i sin c] = R [e^ (+/-)(ic)],
(a, b) = R(cos c, sin c), tan c = b/a,
R = sqrt[a^2 + b^2], R is + ve, irrespective of a>b, or b>a.

Cohy number:
Creating fexponential function, f^x = (cos x + sin x), we write:
Cohy number: a (+/-) ib = H [cosh x (+/-) i sinh x] = H [f^ (+/-)(ix)],
(a, b) = H (cosh x, sinh x), tanh x = b/a,
H = sqrt[a^2-b^2], H is + ve, if a>b, but if b>a, H is complex, say = (iH), (here, H = mod value of H).

Interchangeability of Coci and Cohy parameters:
a^2 + b^2 = R^2 [(cos c)^2 + (sin c)^2] = H^2 [(cosh x)^2 + (sinh x)^2],
so, R = H sqrt[cosh(2x)].
a^2-b^2 = R^2 [(cos c)^2-(sin c)^2] = H^2 [(cosh x)^2-(sinh x)^2],
so, R sqrt[cos(2c)] = H.
From these two, get: cos(2c).cosh(2x) = 1.
Alternative forms are: sin(2c) = tanh(2x), and, tan(2c) = sinh(2x).

For Cohy numbers, special attaintion is drawn, if a<b:
Note that, H is complex, so replace H by (iH), and multiply both sides by (-i).
a + ib = (iH) [cosh x + i sinh x].
(-i)(a + ib) = (-i)(iH) [cosh x + i sinh x].
b-ia = H[cosh x + i sinh x].

Note that since, cosh(-) = (+)ve, sinh(-) = (-)ve,
then write, (b, a) = H[cosh(-x), sinh(-x)]

[The complex number now is, as if (b-ia), and since b>a, the rest of the procedure will follow for the previous case as that for a>b.]

Example-1. Write (8 + 15i) in Coci and Cohy form.
(a, b) = (8, 15), a<b,
(8 + 15i) = R(cos c + sin c) = H(cosh x + sinh x),
R = sqrt[(8^2 + 15^2] = 17,
(cos c, sin c) = (8/17, 15/17), c = 61.9275 deg (=1.0808 rad).
H = sqrt[(8^2-15^2] = 12.6886i (it is complex), /mod H/ = 12.6886.
So, (b-ia) = (15-8i) = H[cosh x + i sinh x],
12.6886 (cosh x, sinh x) = (15, -8), x = -0.5948.
[Taken that, cosh(-) = (+)ve, sinh(-) = (-)ve]

Thus, the Coci and Cohy forms of (8 + 15i), are:
(8 + 15i) = 17e^(1.0808i) = 12.6886f^(-0.5948i)

Check with: tan(2c) = sinh(2x)
sinh(2x) = tan(2c) = -1.49068, 2x = -1.18958, x = -0.5948.

Example-2. (for a>b),
(12 + 5i) = 13e^(0.3948i) = 10.9087f^(0.4436i)
(12-5i) = 13e^(-0.3948i) = 10.9087f^(-0.4436i)

Complex numbers of other forms:
The representation of complex numbers is not restricted to Coci and Cohy varieties, but can be extended to other forms. For example: like the pair (cosh x, sinh x) of hyperbolic functions, pairs of trigonometric (= circular) functions like, (sec u, tan u), (cosec v, cot v) can also be utilized. The method shown for Cohy numbers, will be applicable in these cases, since, [(sec u)^2-(tan u)^2] = 1, and, [(cosec v)^2-(cot v)^2] = 1.

For, (a + ib) = S[sec u + i tan u] = C[cosec v + i cot v], the interchangeability of the parameters are:
(cos 2c).[1 + 2 (tan u)^2] = 1, and, tan c = sin u = cos v
Thanks to All.

Doctor Nisith Bairagi
Uttarpara, West Bengal, India

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