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ARGAND DIAGRAMS OF EXTENDED FIBONACCI AND LUCAS NUMBERS
F, l WUWDERL1CH, D. E. SHAW, and SVL J. HOiES
Department of Physies, Wlllanova Uniwersity, Villanova, Pennsylvania 19085
Numerous extensions of the Fibonacci and Lucas Numbers have been reported in the literature [1-6]. in this
paper we present a computer-generated plot of the complex representation of the Fibonacci and Lucas Numbers.
The complex representation of the Fibonacci Numbers is given by [5,6].
F(x) = <fox - &~x [cos (XTT) + i sin (xn)]
where
0 = LLs/T
F(-x) = i-l)n+1
and
Re[FM]
= -jL- \<t>x-<t>-xcoshx)^
Fix),
;
and
lin[F(x)l
= -j=r { -</>'* sin fax))
V5
The Fibonacci identity: Fix) = Fix - 1) + Fix - 2) is preserved for the complex parts of Fix):
Re [Fix)] = Re [Fix - 1)] + Re [Fix - 2)1
and
lm[F(x)] = lm[F(x - 1)1 + lm[F(x - 2)1.
Figure 1 is a computer-generated Argand plot of Fix) in the range -5 <x < +5.
The branch of the curve for positive x approaches the real axis as x increases. Defining the tangent angle of the
curve as:
{ Re [Fix)] I
this angle approaches zero for large positive x since
f
J?L
=
ImlFM] = 0.
The negative branch of the curve approaches a logarithmic spiral for x large and negative. The modulus r is
given by:
r
= I Re2[Fix)]
in the limit
therefore,
233
+ lm2[F(x)l
)%
234
A R G A i D DIAGRAMS OF EXTENDED FIBONACCI AUB LOCAS NUMBERS
IH
+5
+5
+3 \
•f3 i
+
+S I
-3
-1
+1
+3
OCT. 1074
+5
-3
Fig. 1 Computer-Generated Argand Plot
of the Fibonacci Function
-I
41
+3
+5
Fig. 2 Computer-Generated Argand Plot
of the Lucas Function
l-1/Mk ,
In r
where
and
k = in (<l>A/s)
f\l/k/7T)
r «
-kx
Similarly, the Lucas number identity:
L(x) = Ffx +1) + Fix - 1)
leads directly t o [ 6 ] :
Llx) - 0 X * l-1)x<t>-x
and the complex representation of the Lucas Numbers follows
Llx)
= <px + (p~x (cos rrx + i sin TTX)
with
Re[L(x)]
= <!>* + <j>* cos TTX
Note:
lm[Llx)]
lm[L(x)]
and
-7
<J5
0
x
sin
TTX
.
imlFlx)] .
As with the previous case f o r n large and positive, the positive branch of the Lucas number curve approaches the
Real axis. Again, the negative branch approaches a logarithmic spiral f o r n large and negative.
if ~ m,
r « <f^l/7t)f
Inr « -l^Mln
0,
r ~ e^™*
=
e^x.
REFERENCES
1. W.G. Brady, Problem B-228, The Fibonacci Quarterly, V o l . 10, SMo. 2 (Feb. 1972), p. 218.
2. J.H. Halton, " O n a General Fibonacci I d e n t i t y , " The Fibonacci Quarterly, V o l . 3, No. 1 (Feb. 1965), pp. 3 1 - 4 3 .
3. R . L H e i m e r , " A General Fibonacci F u n c t i o n , " The Fibonacci Quarterly, V o l . 5, No. 5 (Dec. 1967), pp. 4 8 1 - 4 8 3 .
4. E. Halsey, "The Fibonacci Number Fu where u is not an Integer," The Fibonacci Quarterly, Vol. 3, No. 2 (April
1965), pp. 147-152.
5. F. Parker, " A Fibonacci F u n c t i o n , " The Fibonacci Quarterly, V o l . 6, No. 1 (Feb. 1968), pp. 1 - 2 .
6. A.M. Scott, "Continuous Extensions of Fibonacci Identities," The Fibonacci Quarterly, Vol. 6, No. 4 (Oct 1968)
pp. 245-249.
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