已知sinθ cosθ是关于x的方程x^2-ax+a=0的两个根_百度知道已知函数f(x)=-4sin²x+4cosx+1-a,当∈【-π/4,2π/3】时，f(X)=0恒有解，求a的取值范围_百度知道Technical Reference Page
References
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the proper heading on this page -- a link will bring you back.
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Important Formulae for
Complex Numbers
1) z = x + iy where x = Real
part of z and y = Imaginary part of z
2) c = a + ib where a = Real
part of c and b = Imaginary part of c
3) z = re^iq = (sqrt(x^2
+ y^2)) (cos q + i sin q)
where q = arctan (y / x), r = sqrt(x^2 + y^2) and &sqrt& means square
4) z^n = r^n*e^inq = (sqrt(x^2
+ y^2))^n (cos nq + i sin nq) ; r and q as above
5) sqrt(z) = (sqrt(r)sqrt(e^iq))
= (sqrt(sqrt(x^2 + y^2))) [cos (.5 arctan (y / x))
+ i sin (arctan (y / x))]
6) ln z = ln[sqrt(x^2 + y^2)]
+ i arctan (y / x)
7) e^z = e^x(cos y + i sin
8) sin z = sin x cosh y +
i cos x sinh y = -i sinh iz = (e^iz - e^-iz) / 2i
9) cos z = cos x cosh y -
i sin x sinh y = cosh iz = (e^iz + e^-iz) / 2
10) sinh z = - i sinh iz
= (e^z - e^-z) / 2
11) cosh z = cos iz = (e^z
+ e^-z) / 2
12) tanh z = - i tan (iz)
= (e^z - e^-z) / (e^z + e^-z)
13) sech z = sech (iz) =
[cosh z] ^ -1
14) csch z = i csc (iz) =
[sinh z] ^ -1
15) arcsinh z = ln(z + sqrt(z^2
16) arccosh z = ln(z + sqrt(z^2
- 1)) , ln(z - sqrt(z^2 - 1))
17) arctanh z = .5 * ln[(1
+ z) / (1 - z)]
18) arcsech z = ln[(1 + sqrt(z^2
+ 1)) / z]
19) arccsch z = ln[(1 + sqrt(1
- z^2 )) / z] , ln[(1 - sqrt(1 - z^2 )) / z]
20) arccoth z = .5 * ln[(z
+ 1) / (z - 1)]
21) sin^2(z) + cos^2(z) =
22) cosh^2(z) - sinh^2(z)
23) tan z = (sin 2x + i sinh
2y) / (cos 2x + cosh 2y)
24) cot z = (sin 2x - i sinh
2y) / (cosh 2y - cos 2x)
25) nth root of z = [nth
root of (x^2 + y^2)](cos (q / n) + i sin (q / n))
26) Newton's Method z(n+1)
= z(n) - [f(z(n)) / f '(z(n))]
27) Henon Attractor: (for
z(n) = x(n) + iy(n)) , x(n+1) = ax(n) + y(n) and y(n+1)= bx(n)
28) Halley Map: z(n+1) =
z(n) - L[(2f(z(n))f '(z(n))) / (2(f '(z(n)))^2 - f' '(z(n))f(z(n)))]
29) Lorenz Attractor: dx
/ dt = a(y - x) dy / dt = x(r - z) - y dz / dt = xy - bz
Important Mathematical
1) Pi --- The ratio of the
circumference of a circle to its diameter, supposedly first discovered
by Archimedes (287-212 BC). He surmised that pi was
pi & 3 1/7
The first hundred digits
of pi are given here though I understand that 1.24 trillion digits (!) have
been calculated already (there is more about pi on my
Probably the most famous
formula for determining pi is Leibnitz' formula:
pi = 4 - (4/3)
+ (4/5) - (4/7) + (4/9) - (4/11) =
Summation (from
n=0 to infinity) of [(-1)^n][4/(2n+1)]
Another famous summation involving
pi was discovered by Euler as:
(pi^2)/6 = 1/1
+ 1/4 + 1/9 + 1/16 + 1/25 + 1/36 +.... + (1/n)^2
--- The natural logarithm base, supposed named
after the great mathmatician Leonhard Euler. The first hundred digits of e are
given here as well:
For the first 500 digits
of e, go .
I offer my students two ways
to remember how to calculate the value of e:
e = limit (as
n -& infinity) of (1 + 1/n)^n
e = Summation
(from n=0 to infinity) of simply (1/n!)
3) Feigenbaum's Number ---
This number, first shown by Becker and Dorfler, was demonstrated
to be a fundamental constant of nature
having to do with the ratio of intervals of growth rate versus
the doubling of up and down cycles characteristic of that rate.
Keith Briggs, a scientist from the University of Melbourne, Australia,
has calculated the most precise Feigenbaum number to date:
4) Square Root of two = 1.
5) Square Root of three = 1.
6) Square Root of five = 2.
7) Square Root of pi = 1.77245
... (also known as Gamma(.5))
8) Square Root of e = 1. 64872
9) The Golden Mean, phi = (1
+ sqrt(5)) / 2 = 1. 99894...
10) e ^ pi = 23.
11) pi ^ e = 22.
12) e ^ e = 15. 15426
13) Euler's constant (usually
given as lower case gamma) = .
= limit (as
n -& infinity) of (summation (from k=1 to n) of (1/k) - ln n) ---
(thanks to N. Hobson)
14) 1 radian = the number of
degrees that are subtended when the length of a radius is traced
along the circumference of a circle.
1 radian = 180
/ pi = 57.
15) ln 2 = .69315... = 1 -
1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ...
= Summation (from
n = 1 to infinity) of (-1)^(n+1) * (1/n)
16) ln 10 = 2.30259...
There is a library of more
obscure mathematical constants .
Fractal Book References List
(Listed alphabetically as TITLE, AUTHOR, PUBLISHER,
Please submit any important missing
texts to me at .
Advanced Fractal Programming
in C and C++, Stevens, Henry Holt, New York, 1992
An Eye for Fractals: A Graphic
& Photographic Essay, McGuire, Addison-Wesley, Redwood City, CA, 1991
An Introduction to Chaotic
Dynamical Systems, Devaney, Addison-Wesley, New York, 1986
Bifurcation Theory and its
Applications, Kaplan, Yorke,and Williams, New York Academy of Sciences, New
York, 1979
Making A New Science, Gleick, Viking Press, New York, 1987
and the Imagination, Pickover, St. Martin's Press, New York, 1989
Computers,
Pattern, Chaos, and Beauty: Graphics from an Unseen World, Pickover, St. Martin's
Press, New York, 1990
Determistic Chaos, Schuster,
Physik-Verlag, Weinheim, 1984
Does God Play Dice: The Mathematics
of Chaos, Stewart, Basil Blackwell, Cambridge, 1990
Dynamical Systems and Evolution
Equations, Walker, Plenum Press, New York, 1980
Systems and Fractals, Becker and D&rfler, Cambridge Univ. Press, Cambridge,
Exploring the Geometry of
Nature: Computer Modeling of Chaos, Fractals, Cellular Automata, and Neural
Networks, Rietman, Windcrest Books, 1989
Cosmos: The Art of Mathematical Design, Lifesmith, Amber Lotus, Oakland, 1994
Fractal Programming in C,
Stevens, Henry Holt, New York, 1990
Fractal Programming in Turbo
Pascal, Stevens, Henry Holt, New York, 1991
Fractals, Feder, Plenum Press,
New York, 1988
Fractals, Lauwerier, Princeton
univ. Press, Princeton, NJ 1992
Fractals and Multifractals,
Mandelbrot, Springer-Verlag, New York, 1991
Fractals, Chaos, Power Laws,
Schroeder, Freeman, New York, 1991
Everywhere, Barnsley, Academic Press, Boston, 1988
Fractals: Form, Chance, and
Dimension, Mandelbrot, Freeman, San Francisco, 1977
Fractals, The Patterns of
Chaos, Briggs, Touchstone/Simon & Schuster, New York, 1992
FractalVision: Put Fractals
to Work for You, Oliver, Prentice-Hall, Carmel, IN, 1992
Fun with Fractals, Robbins,
Sybex, Alameda, CA 1992
Handbook of Mathematical
Functions, Abramowitz and Stegun, Dover, New York, 1968
Islands of Truth: A Mathematical
Mystery Cruise, Peterson, Freeman, New York, 1990
Iterated Maps on the Interval
as Dynamical Systems, Collet and Ekmann, Birkhauser, Boston, 1980
Mathematics and the Unexpected,
Ekeland, Univ. of Chicago Press, 1988
Order Out of Chaos: Man's
New Dialogue with Nature, Prigogine & Stengers, Bantam, New York, 1984
Symmetry in Chaos, Field
& Golubitsky, Oxford Univ. Press, New York, 1992
Beauty of Fractals, Peitgen and Richter, Springer-Verlag, Berlin, 1986
The Fractal Explorer, Garcia,
Dynamic Press, Santa Cruz, CA, 1991
Fractal Geometry of Nature, Mandelbrot, Freeman, New York, 1980
Selected Papers
Please submit any important
missing papers to me at .
A. K., &Computer Recreations,& Scientific American, Sept. 1986, pp.
P. 1906. Sur les solutions uniformes de certains &quations fonctionelles.
Comptes rendus (Paris) 143, 546-548
Fatou, P. 1919-20. Sur les
&quations fonctionelles. Bull. Soci&t& Math&matique
de France 47, 161-271; 48, 208-314
Feigenbaum, M. J., &Quantitative Universality for a Class
of Nonlinear Transformations,& Journal of Statistical Physics 19, 25-52
Feigenbaum, M. J., &The
Universal Metric Properties of Nonlinear Transformations,& Journal of Statistical
Physics 21, 669-706 (1979)
Hausdorff,
F. 1919. &Dimension und &usseres Mass&. Mathematische Annalen,
79, 157-79
G. 1918. M&moire sur l'it&ration des fonctions rationnelles. J.
de Math&matiques Pures et Appliqu&s 4, 47-245. Reprinted (with
related texts) in Oeuvres de Gaston Julia, Paris, Gauthier-Villars. 1968, p.
E. 1963. &Deterministic Nonperiodic Flows&, Journal of the Atmospheric
Sciences, 20, 130-41
Mandelbrot,
B. B. 1980. Fractal aspects of the iteration of z=lz(1-z) for complex l and
z. Non Linear Dynamics, Ed. R.H.G. Helleman. Annals of New York Academy of Sciences,
357, 249-259
P. F., 1845. R&cherches math&matiques sur la loi d'accroissement
de la population. Nouv. M&m. de l'Acad. Roy. des Sciences et Belles-Lettres
de Bruxelles XVIII. 8, 1-38
R. F. 1985. &Random Fractal Forgeries,& Fundamental Algorithms for
Computer Graph- ics, Springer-Verlag, Berlin
Equations Researched
Here are the equations that
I have used during the past seventeen years to generate well over 400,000 Mandelbrot
and Julia sets. I have over seven TERAbytes of fractal data! Feel free to continue
to delve into them using whatever software (your own or canned) you have available.
Because I wrote my own code in C language and a complex math library was not
available when I first started, I had to resolve each of these equations into
real, f(x), and imaginary, f(y), parts. Many, many long (but fun) hours doing
just the basic algebra were spent in order to bring you the majestic beauty
of these incredible forms.
1--F(Z) = Z^2 + C
2--F(Z) = Z^3 + C
3--F(Z) = (Z^2 + C) / (Z - C)
4--F(Z) = Z^2 - Z + C
5--F(Z) = Z^3 - Z^2 + Z + C
6--F(Z) = (1 + C)Z - CZ^2
7--F(Z) = Z^3 / (1 + CZ^2)
8--F(Z) = (Z - 1)(Z + .5)(Z^2 - 1) + C
9--F(Z) = (Z^2 + 1 + C) / (Z^2 - 1 - C)
10--F(Z) = Z^1.5 + C
11--F(Z) = exp(Z)-C
12--F(Z) = Z - 1 + Cexp(-Z)
13--F(Z) = CZ - 1 + Cexp(-Z)
14--F(Z) = (4Z^5 + C)/5Z^4
15--F(Z) = (6Z^7 + C)/7Z^6
16--F(Z) = Z^2 * exp(-Z) + C
17--F(Z) = Z^2 * Z^(-2) + C
18--F(Z) = Z * exp(-Z) + C
19--F(Z) = C * exp(-Z) + Z^2
20--F(Z) = Z^3 + Z + C
21--F(Z) = Z^4 + Z + C
22--F(Z) = Z^4 + CZ^2 + C
23--F(Z) = Z^2sin(Re Z) + CZcos(Im Z) + C
24--F(Z) = 2^Z * CZ^2
25--F(Z) = Z^5 - Z^3 + Z + C
26--F(Z) = (Z^2 + C)^2 + Z + C
27--F(Z) = (Z + sin(Z))^2 + C
28--F(Z) = Cexp(Z)
29--F(Z) = Z^2 + C^3
30--F(Z) = Cexp(CZ)
31--F(Z) = Z^2cos(ReZ)+CZsin(ImZ)+C
32--F(Z) = CZ^2 + ZC^2
33--F(Z) = exp(cos(CZ))
34--F(Z) =(1 + Jo(Re Z))^2 + (Jo(Im Z) + C)^2
(Here Jo represents the Bessel function)
35--F(Z) = C(sin Z + cos Z)
36--F(Z) = Z^(-.5) + C
37--F(Z) = CZ(1 - Z)
38--F(Z) = C^2Z(1 - Z)
39--F(Z) = ((Z^2+C)^2)/(Z-C)
40--F(Z) = (Z + sin Z)^2 + Z^-.5 + C
41--F(Z) = C*(sin Z + cos Z)*(Z^3+Z+C)
42--F(Z) = Cexp(Z) * exp(cosCZ)
43--F(Z) = (Z^3+Z+C)*C*(sinZ + cosZ)
44--F(Z) = ((1+C)Z-CZ^2)*((Z+sinZ)^2+C)
45--F(Z) = Z^2 + Z^1.5 + C
46--F(Z) = Z^2 + ZexpZ + C
47--F(Z) = (Z+sinZ)^2+Cexp(-Z)+Z^2+C
48--F(Z) = ((Z^3)/(1+CZ^2))+expZ-C
49--F(Z) = (Z^2*sin(ReZ) + CZ(ImZ) + (Z^2*cos(ReZ)+CZsin(ImZ)+C
50--F(Z) = (Z+sinZ)^2+Cexp(Z)+C
51-- F(Z) = Z^2 + 1/Z + C
52-- F(Z) = (Z^3 + C) / Z
53-- F(Z) = (Z^3 + C) / Z^2
54-- F(Z) = ((Z+1)^2 + C) / Z
55-- F(Z) = (Z + C)^2 + (Z + C)*
56-- F(Z) = (Z + C)^3 - (Z + C)^2
57-- F(Z) = (Z^3 - Z^2)^2 + C
58-- F(Z) = (Z^2 - Z)^2 + C
59-- F(Z) = (Z + ln Z)^2 + C
60-- F(Z) = (Z - sqrt(Z))^2 + C
61-- F(Z) = (Z + sqrt(Z))^2 + C
62-- F(Z) = Z^2exp(Z) - Zexp(Z) + C
63-- F(Z) = (exp(CZ) + C)^2
64-- F(Z) = Z * exp(Re Z/Im Z) + C
65-- F(Z) = exp(X^2*Y^2) + Im Z + C
66-- F(Z) = exp(Re Z)*(X-a) + exp(Im Z)*(Y-b)i
67-- F(Z) = X^2*exp(Y+b) + iaexp(Y+b)
68-- F(Z) = (a-X^2+Y^2)exp(b+X^2-Y^2) + i(b+X^2-Y^2)exp(a-X^2+Y^2)
69-- F(Z) = [(2X-Y^2+a)/(2X^2+Y-b)] + i[(2X^2+Y-a)/(2X-Y^2+b)]
70-- F(Z) = [(X^2+Y^2+a)/cos(X^2+Y^2)] + i[(X^2+Y^2+b)/sin(X^2+Y^2)]
71-- F(Z) = Z^5 + C
72-- F(Z) = Z^6 + C
73-- F(Z) = Z^7 + C
74-- F(Z) = (3Z^4 + C) / 4Z^3
75-- F(Z) = (2Z^3 + C) / 3Z^2
76-- F(Z) = Z^5 + CZ^3 + C
77-- F(Z) = Z^6 + CZ^4 + CZ^2 + C
78-- F(Z) = Z^8 + C
79-- F(Z) = Z^9 + C
80-- F(Z) = Z^8 + CZ^4 + CZ^2 + C
81-- F(Z) = Z^9 - CZ^6 + CZ^3 + C
82-- F(Z) = (Z^4 + C) / (Z - C)
83-- F(Z) = (Z^3 + Z + C) / (Z^2 - Z - C)
84-- F(Z) = (Z^3 + Z + C) / (Z - C)
85-- F(Z) = (Z^3 + Z + C) / Z
86-- X = X^2+XY+A ; Y = Y^2-XY+B
87-- X = X^3-(X^2)Y+XY^2-XY+A; Y = Y^3-XY^2+(X^2)Y+XY+B
88-- X = (X^2)sin Y + A ; Y = (Y^2)cos X + B
89-- X = X^4-3X^3+3X^2(Y^2)+A ; Y = Y^4+3XY^3-3X^2(Y^2)
90-- X = X^2(1+exp(-Y))+A ; Y = Y^2(1+exp(-X)+B
91-- F(Z) = C(Z^2 + 1)^2 / Z(Z^2 -1)
92-- F(Z) = CZ^2
93-- F(Z) = CZ^3
94-- F(Z) = CZ^4
95-- F(Z) = C*cos Z
96-- F(Z) = C*sin Z
97-- F(Z) = CZ*ln Z
98-- F(Z) = C*tan Z
99-- F(Z) = C*exp(CZ) / (exp(C) - 1)
100-- F(Z) = C*exp(Z)*sqrt(Z) /n
101 -- F(Z) = (Z^2(1+Z^2))/(Z+C)
102 -- F(Z) = Z(1+Z^2)/(Z+C)
103 -- F(Z) = (Z^5+C)/(Z^3+Z^2+Z+1)
104 -- F(Z) = (Z^3+C)/3Z^2
105 -- F(Z) = (Z^3+Z^2+Z+C)/(Z-C)
106 -- F(Z) = exp(Z^2+C)
107 -- F(Z) = Z^2*exp(Z^2)+C
108 -- F(Z) = exp(Z^2)/(Z+C)
109 -- F(Z) = (Z+exp(Z))^2+C
110 -- F(Z) = (Z^2+C)^2-exp(Z)+C
111 -- F(Z) = (1+iC)sin(Z)
112 -- F(Z) = (1+iC)cos(Z)
113 -- F(Z) = Z*tan(ln Z)+C
114 -- F(Z) = sqrt(Z^4+1)+C
115 -- F(Z) = sqrt(Z^4+C)
116 -- F(Z) = C^Z
117 -- F(Z) = C*arctan(Z)
118 -- F(Z) = (ZlnZ)/exp(C)
119 -- F(Z) = exp(Z)/lnZ+C
120 -- F(Z) = sqrt(Z^3+C)
121 -- F(Z) = sqrt(Z^3+1)+C
122 -- F(Z) = cubrt(Z^6+1)+C
123 -- F(Z) = (Z+exp(Z)+ln Z)^2+C
124 -- F(Z) = (Z^2+C+1)^2 / (2Z+C+2)^2
125 -- F(Z) = Z ^ 10 + C
126 -- F(Z) = Z ^ 11 + C
127 -- F(Z) = Z ^ 12 + C
128 -- F(Z) = Z^12 - Z^11 - Z^10 + C
129 -- F(Z) = Z ^ 13 + C
130 -- F(Z) = Z ^ 14 + C
131 -- F(Z) = Z ^ 15 + C
132 -- F(Z) = Z ^ 16 + C
133 -- F(Z) = Z ^ 17 + C
134 -- F(Z) = Z ^ 18 + C
135 -- F(Z) = Z ^ 19 + C
136 -- F(Z) = Z ^ 20 + C
137 -- F(Z) = Z ^ 21 + C
138 -- F(Z) = Z ^ 22 + C
139 -- F(Z) = Z ^ 23 + C
140 -- F(Z) = Z ^ 24 + C
141 -- F(Z) = Z ^ 25 + C
142 -- F(Z) = Z ^ 26 + C
143 -- F(Z) = Z ^ 27 + C
144 -- F(Z) = Z ^ 28 + C
145 -- F(Z) = Z ^ 29 + C
146 -- F(Z) = Z^30 + C
147 -- X=X^2+Y+A+X^2/Y ;Y=Y^2+X+B+Y^2/X
148 -- X=X^3+Y^2-X+A ;Y=Y^3-X^2+Y+B
149 -- X=X^2+2XY-Y+A ;Y=Y^2-2XY+X+B
150 -- X=X^3+AX^2+BY ;Y=Y^3+BY^2+AX
151 -- X=2X^2-3ABY+A ;Y=3Y^2+2ABX-B
152 -- X=X^4lnX+Y^2sinY+A; Y=Y^4lnY+X^2cosX+B
153 -- X=sqr(ln(X^2))+YsinX+A; Y=sqr(ln(Y^2))-XcosY+B
154 -- X=.5(X^2-Y^2)+.5(X+Y)+A; Y=.5(Y^2-X^2)-.5(X+Y)+B
155 -- X=sqr(X^3)+sqr(Y^3)+A; Y=sqr(Y^3)-sqr(X^3)+B
156 -- X=Y/sqrX+X/sqrY+A; Y=XsqrY+YsqrX+B
157 -- F(Z) = Z^30 - 30Z^29 - 870Z^28 + C
158 -- F(Z) = Z^27 - 27Z^26 - 702Z^25 + C
159 -- F(Z) = Z^24 - 24Z^23 - 552Z^22 + C
160 -- F(Z) = Z^21 - 21Z^20 - 420Z^19 + C
161 -- F(Z) = Z^18 - 18Z^17 - 306Z^16 + C
162 -- F(Z) = Z^15 - 15Z^14 - 210Z^13 + C
163 -- F(Z) = Z^30 - 30Z^29 - 870Z^28 + Z^27
- 27Z^26 - 702Z^25 + C
164 -- F(Z) = Z^27 - 27Z^26 - 702Z^25 + Z^24
- 24Z^23 - 552Z^22 + C
165 -- F(Z) = Z^24 - 24Z^23 - 552Z^22 + Z^21
- 21Z^20 - 420Z^19 + C
166 -- F(Z) = Z^21 - 21Z^20 - 420Z^19 + Z^18
- 18Z^17 - 306Z^16 + C
167 -- F(Z) = Z^18 - 18Z^17 - 306Z^16 + Z^15
- 15Z^14 - 210Z^13 + C
168 -- F(Z) = Z^30 - 30Z^29 - 870Z^28 + Z^27
- 27Z^26 - 702Z^25 + Z^24 - 24Z^23 -
552Z^22 + C
169 -- F(Z) = Z^27 - 27Z^26 - 702Z^25 + Z^24
- 24Z^23 - 552Z^22 + Z^21 - 21Z^20 -
420Z^19 + C
170 -- F(Z) = Z^24 - 24Z^23 - 552Z^22 + Z^21
- 21Z^20 - 420Z^19 + Z^18 - 18Z^17 -
306Z^16 + C
171 -- F(Z) = Z^21 - 21Z^20 - 420Z^19 + Z^18
- 18Z^17 - 306Z^16 + Z^15 - 15Z^14 -
210Z^13 + C
172 -- F(Z) = Z^30 - Z^29 + Z^28 - Z^27 + Z^26
- Z^25 + C
173 -- F(Z) = Z^24 - Z^23 + Z^22 - Z^21 + Z^20
- Z^19 + C
174 -- F(Z) = Z^18 - Z^17 + Z^16 - Z^15 + Z^14
- Z^13 + C
175 -- F(Z) = Z^15sinX - Z^14cosY - Z^13tanX
176 -- F(Z) = Z^12cosX - Z^11sinY - Z^10tanY
177 -- F(Z) = Z^15sinA - Z^14cosB - Z^13tanX
- Z^12tanY + C
178 -- F(Z) = Z^12cosA - Z^11sinB - Z^10tanY
- Z^9tanX + C
179 -- F(Z) = Z^30sinX - 30Z^29cosY + C
180 -- F(Z) = Z^28cosX - 28Z^27sinY + C
181 -- F(Z) = (Z^3+3Z(C-1)+(C-1)(C-2))^2
182 -- F(Z) = (3Z^2+3Z(C-2)+C^2-3C+3)^2
183 -- F(Z) = (Z^3+3Z(C-1)+(C-1)(C-2))^2 / (3Z^2+3Z(C-2)+C^2-3C+3)^2
184 -- F(Z) = Z ^ pi + C
185 -- F(Z) = pi ^ Z + C
186 -- F(Z) = Z ^ 4 + C
187 -- F(Z) = Z ^ pi + pi ^ C
188 -- F(Z) = C * Z ^ pi
189 -- F(Z) = Z ^ pi - Z ^ 3 + C
190 -- F(Z) = Z ^ pi - Z ^ 2 + C
191 -- F(Z) = Z ^ 2.5 + C
192 -- F(Z) = (5Z^6 + C)/6Z^5
193 -- F(Z) = Z ^ e + C
194 -- F(Z) = Z ^ (C * e)
195 -- F(Z) = (Z ^ e) ^ C
196 -- F(Z) = C * Z ^ e
197 -- F(Z) = Z ^ (pi * e)
198 -- F(Z) = Z * (C ^ e)
199 -- F(Z) = cbrt(Z ^ 7 + 1) + C
200 -- F(Z) = Z ^ 4.669 + C
201 -- F(Z) = (Z ^ 8 + 1) ^ 1/4 + C
202 -- F(Z) = (Z ^ 9 + 1) ^ 1/4 + C
203 -- F(Z) = ((Z ^ 2 * (ReZ - (ImZ)^2))/(1
204 -- F(Z) = (Z ^ 10 + C) ^ 1/4
205 -- F(Z) = (Z ^ 10 + 1) ^ 1/4 + C
206 -- F(Z) = (Z ^ 11 + C) ^ 1/4
207 -- F(Z) = (Z ^ 11 + 1) ^ 1/4 + C
208 -- F(Z) = (Z ^ 12 + C) ^ 1/4
209 -- F(Z) = (Z ^ 12 + 1) ^ 1/4 + C
210 -- F(Z) = YZ^2sinX - XZcosY + C
211 -- F(Z) = XZ^3cosY + YZ^2sinX + C
212 -- F(Z) = Z^4 - Z^2cosX + YsinY + C
213 -- F(Z) = XYZ^2 + C
214 -- F(Z) = Z^2 + X^2*Y^2 + C
215 -- F(Z) = Z^3 + X^2sinY + Y^2cosX + C
216 -- F(Z) = (Z ^ 13 + C) ^ 1/6
217 -- F(Z) = (Z ^ 5 + C) ^ 1/3
218 -- F(Z) = (Z ^ 4 + C) ^ 1/sin X
219 -- F(Z) = Z ^ 2 + iZ ^ 2 + C
220 -- F(Z) = Z ^ 3 + iZ ^ 3 + C
221 -- F(Z) = Z ^ 4 + iZ ^ 2 + C
222 -- F(Z) = (Z ^ 4 / Z + 1) + C
223 -- F(Z) = (Z ^ 6 / Z + 1) + C
224 -- F(Z) = (Z ^ 4 / Z + i) + C
225 -- F(Z) = (Z ^ 6 / Z + i) + C
226 -- F(Z) = (Z ^ 2 / (lnZ)^2) + C
227 -- F(Z) = (Z ^ 2 / (ln(Z^2)) + C
228 -- F(Z) = (Z ^ 3 / (lnZ)^3) + C
229 -- F(Z) = (Z ^ 3 / (ln(Z^3)) + C
230 -- F(Z) = (Z ^ 4 / (lnZ)^4) + C
231 -- F(Z) = (Z ^ 4 / (ln(Z^4)) + C
232 -- F(Z) = Z ^ 2 + Z / ln Z + C
233 -- F(Z) = Z ^ 2 + ln Z / Z + C
234 -- F(Z) = Z ^ 6 + Z ^ 4 + Z ^ 2 + C
235 -- F(Z) = Z ^ 6 - Z ^ 4 - Z ^ 2 + C
236 -- F(Z) = Z ^ (1/Z) + C
237 -- F(Z) = Z ^ 2 + sin Z / Z + C
238 -- F(Z) = Z ^ 2 + Z / sin Z + C
239 -- F(Z) = Z ^ iZ + C
240 -- F(Z) = Z ^ 2 * exp(X) + C
241 -- F(Z) = Z ^ 2 * exp(X ^ 2) + C
242 -- F(Z) = Z ^ 3 * exp(X) + Z ^ 2 * exp(Y)
243 -- F(Z) = exp(Z ^ Z) + C
244 -- F(Z) = (Z ^ 3) / (Z + 1) + C
245 -- F(Z) = Z ^ 2 / C
246 -- F(Z) = (Z ^ 4 + 1) / (Z + C)
247 -- F(Z) = (Z ^ 4 + C) / (Z ^ 2 + 1)
248 -- F(Z) = (Z ^ 4 + C) / (1 - Z ^ 2)
249 -- F(Z) = Z ^ 2 * exp(Z) / (Z + C)
250 -- F(Z) = Z ^ 2 - exp(Z) + sin(Z) + C
251 -- F(Z) = (Z ^ 4) / (Z ^ 2 + C)
252 -- F(Z) = Z ^ 2 + sqrt(Z ^ 2 + C)
253 -- F(X) = X^2 - Y^2 + XsinY + A; F(Y) =
254 -- F(X) = X^2 + atan(Y/X) + A; F(Y) = Y^2
255 -- F(X) = 1 - X - Y^2 + A; F(Y) = 1 - Y
256 -- F(X) = exp(sqrt(X)) - exp(sqrt(Y)) +
A; F(Y) = exp(XlnY) + B
257 -- F(Z) = C ^ 2 * ln(Z ^ 2)
258 -- F(Z) = Z ^ 2 ln(C)
259 -- F(Z) = Z ^ 2 ln(C) + C
260 -- F(Z) = Z ^ 2 ln(Z + C)
261 -- F(Z) = Z ^ -2 + C
262 -- F(Z) = ((X^2 + Y^2 + A) / (X^2 - Y^2))
+ i[((X^2 - Y^2 - B) / (X^2 + Y^2))]
263 -- F(Z) = (X^3 - iY + C) / (X + Y + 1)
264 -- F(Z) = [(X^2 + A^2) / Y] + i[(Y^2 + B^2)
265 -- F(Z) = [(X^3 + X^2 + X + A) / (Y^3 -
Y^2 - Y - 1)] + i[(Y^3 + Y^2 + Y + B) /
(X^3 - X^2 - X - 1)]
266 -- F(Z) = [(X^4 - Y^2) / (X + Y + A)] +
i[(X^2 + Y^4) / (X - Y - B)]
267 -- F(Z) = C ^ 3 / Z ^ 2
268 -- F(Z) = [Z^(1/2) / Z^(1/3)] + C
269 -- F(Z) = (Z ^ 2 + C) / (1 - C)
270 -- F(Z) = (exp(Z ^ 4)/ Z ^ 4) + C
271 -- F(Z) = (Z ^ 6 + 1) ^ (1/5) + C
272 -- F(Z) = Z ^ 2 + CZ + C * sin Y - Z * cos
273 -- F(Z) = Z ^ 6 - Z ^ 5 - Z ^ 4 - Z ^ 3
- Z ^ 2 - Z + C
274 -- F(Z) = Z ^ 2 * (sin C / C)
275 -- F(Z) = exp(- Z ^ 2 / 2) + C
276 -- F(Z) = (Z ^ 3 + 3 * Z - 1) / (2 - Z)
277 -- F(Z) = Z ^ 3 * sin C + Z ^ 2 * cos C
278 -- F(Z) = (Z ^ 2 + 1) ^ 2 / (Z + C) ^ 2
279 -- F(Z) = (Z ^ 2 + C + 1) ^ 2 / (Z - C -
280 -- F(Z) = Z ^ 4 * sin Y + Z ^ 2 * cos X
281 -- F(Z) = Z * cos (XY) + C
282 -- F(Z) = Z ^ 2 * cos (X ^ 2 + Y ^ 2) +
283 -- F(Z) = Z ^ (2 + ln C)
284 -- F(Z) = Z ^ (9/7) + C
285 -- F(Z) = Z ^ 5 * (1 - Z - (Z + C) ^ 2)
286 -- F(Z) = Z ^ 6 + Z ^ 5 + C
287 -- F(Z) = Z ^ 6 + Z ^ 4 + C
288 -- F(Z) = Z ^ 6 + Z ^ 3 + C
289 -- F(Z) = Z ^ 6 + Z ^ 2 + C
290 -- F(Z) = Z ^ 6 + Z + C
291 -- F(Z) = Z ^ 2 + cos Z + C
292 -- F(Z) = Z ^ 2 + cos 2Z + C
293 -- F(Z) = Z ^ 2 + cos 3Z + C
294 -- F(Z) = Z ^ 2 + cos 4Z + C
295 -- F(Z) = Z ^ 2 + cos 5Z + C
296 -- F(Z) = (Z ^ 7 + C) / Z ^ 5
297 -- F(Z) = (Z ^ 7 + C) / Z ^ 4
298 -- F(Z) = (Z ^ 7 + C) / Z ^ 3
299 -- F(Z) = (Z ^ 7 + C) / Z ^ 2
300 -- F(Z) = (Z ^ 7 + C) / Z
301 -- F(Z) = Z ^ 3 - Z ^ 2 - Z + C
302 -- F(Z) = Z ^ 4 - Z ^ 3 - Z ^ 2 + C
303 -- F(Z) = Z ^ 5 - Z ^ 4 - Z ^ 3 + C
304 -- F(Z) = Z ^ 6 - Z ^ 5 - Z ^ 4 + C
305 -- F(Z) = Z ^ 7 - Z ^ 6 - Z ^ 5 + C
306 -- F(Z) = Z ^ 2 * (cos(Z)) ^ 2 + C
307 -- F(Z) = Z ^ 2 * (cos(XY)) ^ 2 + C
308 -- F(Z) = Z ^ 4 * (sin(Z)) ^ 2 + C
309 -- F(Z) = Z ^ 3 * (sin(XY)) ^ 2 + C
310 -- F(Z) = Z ^ 3 * (cos(Z)*sin(Z)) + C
311 -- F(Z) = (Z ^ 2 / sin(Z)) + C
312 -- F(Z) = (Z ^ 4 / cos(Z)) + C
313 -- F(Z) = (Z ^ 6 + C) / (sin(Z) * cos(Z))
314 -- F(Z) = (Z ^ 3 + Z ^ 2 + Z + C) / (Z +
315 -- F(Z) = (Z ^ 2 * ln Z + Z + C) / (sin(Z))
316 -- F(Z) = Z ^ 4 + (cos X) ^ 2 + (sin Y)
317 -- F(Z) = Z ^ 3 + cos X * sin Y + C
318 -- F(Z) = Z ^ 4 + Z + cos C
319 -- F(Z) = Z ^ 2 + Z + tan C
320 -- F(Z) = Z ^ 3 + Z ^ 2 + exp(1 + sin X)
321 -- F(Z) = sqrt(Z ^ 4 + cos(theta) + C);
theta = arctan (Im Z / Re Z)
322 -- F(Z) = sqrt(Z ^ 5 + Z ^ 3 + Z + C)
323 -- F(Z) = sqrt(Z ^ 4 + Z ^ 3 + Z ^ 2 + Z
324 -- F(Z) = sqrt(Z ^ 6 - Z ^ 3 + C)
325 -- F(Z) = sqrt(ln (Z ^ 2) + Z ^ 2 * ln Z
326 -- F(Z) = cos((Z ^ 2 + C) / XY)
327 -- F(Z) = cos((Z ^ 3 + C) / XY)
328 -- F(Z) = cos((Z ^ 4 + C) / XY)
329 -- F(Z) = ((Z ^ 4 + C) / XY) + cos((Z ^
3 + C) / XY)
330 -- F(Z) = cos((Z ^ 4 + C) / XY) + cos((Z
^ 3 + C) / XY) + cos((Z ^ 2 + C) / XY)
331 -- F(Z) = Z ^ 3/2 + Z ^ 4/3 + C
332 -- F(Z) = Z ^ 4/3 + Z ^ 5/4 + C
333 -- F(Z) = Z ^ 5/4 + Z ^ 6/5 + C
334 -- F(Z) = Z ^ 5/2 + Z ^ 7/3 + C
335 -- F(Z) = Z ^ (pi/e) ^ 2 + C
336 -- F(Z) = Y * sin X * cos Y * exp(-X) +
337 -- F(Z) = Z ^ 2 * cos X * cos Y * exp(-Y)
338 -- F(Z) = XYZ * sin X * sin Y * exp(Z) +
339 -- F(Z) = Z ^ 3 + X ^ 2 * Y ^ 2 * cos X
* sin Y + C
340 -- F(Z) = Z ^ 2 + X ^ 2 * sin Y + Y ^ 2
* cos X + C
341 -- F(Z) = 1 / Z + 1 / Z ^ 2 + C
342 -- F(Z) = Z ^ 2 / Z' + Z ^ 3 / Z' ^ 2 +
343 -- F(Z) = Z ^ 3 / C' + Z ^ 2 + C
344 -- F(Z) = Z ^ 2 + Z' ^ 2 + C
345 -- F(Z) = Z ^ 3 + Z ^ 2 * Z' + Z * Z' ^
2 + Z' ^ 3 + C
346 -- F(Z) = Z ^ 4 - Z ^ 3 * Z' + Z ^ 2 - Z'
347 -- F(Z) = Z ^ 5 - C * Z ^ 3 - C' * Z ^ 2
348 -- F(Z) = Z ^ 4 * Z' ^ 2 - Z ^ 3 * Z' +
349 -- F(Z) = Z ^ 6 + Z' ^ 5 + Z ^ 4 + Z' ^
3 + Z ^ 2 + Z' + C
350 -- F(Z) = Z ^ 4 + Z ^ 2 / Z' + Z' ^ 3 /
351 -- F(Z) = arcsin(ln(Z)) + C
352 -- F(Z) = arctan(ln(Z)) + C
353 -- F(Z) = (arcsin(ln(Z))) ^ 2 + C
354 -- F(Z) = e ^ (1 + cos(ln(Z))) + C
355 -- F(Z) = e ^ (2 - e ^ cos(Z)) + C
356 -- F(Z) = XY * Z^2 - X^2 * YZ + X * Y ^
2 * Z ^ 3 + C
357 -- F(Z) = X ^ 3 * Y ^ 4 + X ^ 2 * Z ^ 5
358 -- F(Z) = XY^2Z^3 - X^3Y^2Z + X^2Y^2Z^2
359 -- F(Z) = Z ^ 4 - X ^ 2 * cos(Y) + Y * sin(X)
360 -- F(Z) = Z ^ 3 - Y ^ 2 * cos(XY) - X ^
2 * sin(X) - Y * cos(Y) + C
361 -- F(Z) = (Z ^ 3 + C) / (Z ^ 3 - C)
362 -- F(Z) = (Z ^ 3 + C ^ 2 + 1) / (Z ^ 3 -
C ^ 2 - 1)
363 -- F(Z) = (Z ^ 3 + Z + C) / (Z ^ 3 - Z -
364 -- F(Z) = (Z ^ 2 - Z ^ 3 + 1) / (Z ^ 4 +
365 -- F(Z) = (Z ^ 4 + C) / (4Z ^ 3 + 1)
366 -- F(Z) = (Z ^ C) / (Z + 1)
367 -- F(Z) = (Z ^ (1 + C)) / (1 + C)
368 -- F(Z) = (2 ^ Z) / C
369 -- F(Z) = 2 ^ Z + C
370 -- F(Z) = 2 ^ Z + (2 ^ Z) / C + C
371 -- F(Z) = XYZ ^ 2 - X ^ 2YZ + C
372 -- F(Z) = X ^ 4 * Y ^ 3 * Z ^ 2 + C
373 -- F(Z) = X^3*Y^3*Z^3 - X^2*Y^2*Z^2 + XYZ
374 -- F(Z) = XY^2Z + X^2YZ^4 + C
375 -- F(Z) = Z^2*sqrt(XY) + XY^2Z^4*sqrt(XY)
376 -- F(Z) = Z ^ (2XY) + C
377 -- F(Z) = X ^ (2YZ) + C
378 -- F(Z) = Y ^ (Z^2) + X + C
379 -- F(Z) = X ^ (2YZ) + Z ^ (2XY) + C
380 -- F(Z) = (XY) ^ (Z - C)
381 -- F(Z) = Z ^ 2C + C
382 -- F(Z) = Z ^ 2 + C ^ 2Z + C
383 -- F(Z) = X^Y + Z^X + Y^Z + C
384 -- F(Z) = Z ^ 2 + A ^ X + B ^ Y + C
385 -- F(Z) = Z ^ 3 + X ^ (AB) + Y ^ C
386 -- F(Z) = Z ^ 9 - Z ^ 8 - Z ^ 7 + C
387 -- F(Z) = Z^9 - 9Z^8 - 8Z^7 + C
388 -- F(Z) = Z^9 - 9Z^8 - 8Z^7 - 7Z^6 - 6Z^5
- 5Z^4 + C
389 -- F(Z) = Z^9 - 9Z^8 - 8Z^7 - 7Z^6 - 6Z^5
- 5Z^4 - 4Z^3 - 3Z^2 - 2Z +C
390 -- F(Z) = Z^9 + C^9
391 -- F(Z) = Z^3 + Z^2 + CsinX + C
392 -- F(Z) = Z^4 + X^2 - Y^2 - C^2*cosZ + C
393 -- F(Z) = Z^5 + Im(Z^4 + Z^3 + Z^2) + CZRe(Z^2
394 -- F(Z) = Z^3 + Z^2*cosY + ZsinX + C
395 -- F(Z) = Z^4 + (Z^2 / sinY) + (CZ^3 / cosX)
396 -- F(Z) = (Z + ln Z)^4 + C
397 -- F(Z) = Z + (ln Z)^4 + C
398 -- F(Z) = Z^2 + (ln Z)^3 + C
399 -- F(Z) = Z^3 + (ln Z)^2 + C
400 -- F(Z) = (ln Z)^2 + C^2 + C
401 -- F(Z) = Z^2 + C + A
402 -- F(Z) = Z^2 + C + iB
403 -- F(Z) = Z^2 + C + X
404 -- F(Z) = Z^2 + C + iY
405 -- F(Z) = Z^2 + C + iXY
406 -- F(Z) = X^3 + X^2Y - XY^2 + Y^3 + C
407 -- F(Z) = X^5 + iX^4 -iX^3Y + iX^2Y^2 - iXY^3 + iY^4 + C
408 -- F(Z) = Y^3 - Y^2 - Y - 1 - A + iX^3 + iX^2 + iX + i + iB
409 -- F(Z) = Y^4 + Y^2 + A - iX^4 - iX^2 - iB
410 -- F(Z) = X^2 + X + 1 + A + iY^2 - iY - i - iB
411 -- F(Z) = arcsin(Z) + C (3rd appr)
412 -- F(Z) = 1 - Z/X - Z^2/Y + C
413 -- F(Z) = arctan(Z) + C (5th appr)
414 -- F(Z) = X^3/Y^2 + Y^4/X^3 + A + i(Y^2/X) - i(X^3/Y) + iB
415 -- F(Z) = Z^4/(X+Y) + Z^3/(X-Y) + C
416 -- F(Z) = Z^(2-X) + C
417 -- F(Z) = Z^(2-X-Y) + C
418 -- F(Z) = Z^(3+C)
419 -- F(Z) = Z^(2X^2 - 3Y^2) + C
420 -- F(Z) = (X+Y)^2Z + C
421 -- F(Z) = Z^2 + sin(Z)*cos(Z) + C
422 -- F(Z) = Z^2 + sec(Z)*tan(Z) + C
423 -- F(Z) = Z^2 + sin(Z)*tan(Z) + C
424 -- F(Z) = Z^2 + cot(Z)*arcsin(Z)
425 -- F(Z) = Z^2 + tan(Z)*arccos(Z) + C
426 -- F(Z) = Z^(2-Z) + C
427 -- F(Z) = Z^(3-Z) + C
428 -- F(Z) = Z^(4-Z) + C
429 -- F(Z) = Z^(5-Z) + C
430 -- F(Z) = Z^(6-Z) + C
431 -- F(Z) = Z^(2-C) + C
432 -- F(Z) = Z^(3-C) + C
433 -- F(Z) = Z^(4-C) + C
434 -- F(Z) = Z^(5-C) + C
435 -- F(Z) = Z^(6-C) + C
436 -- F(Z) = Z^Z / Z^C
437 -- F(Z) = C^Z / Z^Z
438 -- F(Z) = (1-C)^2Z
439 -- F(Z) = (Z+C)^Z
440 -- F(Z) = (Z^2 + C^2)^2Z
441 -- F(Z) = Z^CZ
442 -- F(Z) = Z^(C^2) + C
443 -- F(Z) = C^(2^Z) + C
444 -- F(Z) = Z^(tan(Z) + C
445 -- F(Z) = Z^(sin(Z)cos(Z)) + C
446 -- F(Z) = 2CZ / (Z + C)
447 -- F(Z) = 2CZ / sqrt(Z + C)
448 -- F(Z) = 2 * Z^2 * C^2 / (Z + C)
449 -- F(Z) = 2CZ^3 / (Z + C)
450 -- F(Z) = sqrt(2CZ) / (Z + C)
451 -- F(Z) = Z ^ (Z/C)
452 -- F(Z) = Z^2 + Z ^ (Z/C)
453 -- F(Z) = Z ^ (Z/C) + Z + C
454 -- F(Z) = Z ^ (Z/C) Z ^ 2 + C
455 -- F(Z) = Z ^ (Z^2 / C)
456 -- F(Z) = Z ^ (C/Z)
457 -- F(Z) = Z ^ (C/Z) + Z
458 -- F(Z) = Z ^ (C/Z) + Z^2
459 -- F(Z) = Z ^ (C/Z) + C
460 -- F(Z) = Z ^ (C^2 / Z)
461 -- F(Z) = Z ^ ln(Z ^ Z/C)
462 -- F(Z) = Z ^ ln(Z ^ C/Z)
463 -- F(Z) = Z ^ Z/C + Z ^ C/Z
464 -- F(Z) = sqrt(X^2 + Y^2) + iarctan(Y/X) + C
465 -- F(Z) = Z^2 + sqrt(X^2 + Y^2) + iarctan(Y/X) + C
466 -- F(Z) = Z^3 + sqrt(X^2 + Y^2) + iarctan(Y/X) + C
467 -- F(Z) = Z^4 + sqrt(X^2 + Y^2) + iarctan(Y/X) + C
468 -- F(Z) = Z^2 + sqrt(A^2 + B^2) + iarctan(B/A)
469 -- F(Z) = Z^3 + sqrt(X^3 + Y^3) + C
470 -- F(Z) = Z^4 + sqrt(X^4 + Y^4) + C
471 -- F(Z) = Z^5 + sqrt(X^5 + Y^5) + C
472 -- F(Z) = Z^6 + sqrt(X^6 + Y^6) + C
473 -- F(Z) = Z^7 + sqrt(X^7 + Y^7) + C
474 -- F(Z) = Z^8 + sqrt(X^8 + Y^8) + C
475 -- F(Z) = Z^9 + sqrt(X^9 + Y^9) + C
476 -- F(Z) = sqrt(X^2 + Y^2) + isqrt(X^3 + Y^3)
477 -- F(Z) = sqrt(X^3 + Y^3) + isqrt(X^4 + Y^4)
478 -- F(Z) = sqrt(X^4 + Y^4) + isqrt(X^5 + Y^5)
479 -- F(Z) = sqrt(X^5 + Y^5) + isqrt(X^6 + Y^6)
480 -- F(Z) = sqrt(X^6 + Y^6) + isqrt(X^7 + Y^7)
481 -- F(Z) = Z^2 + exp(cot(Z))*cot(Z)
482 -- F(Z) = Z^2 + exp(tan(Z))*tan(Z)
483 -- F(Z) = Z^2 + exp(cos(Z))*cos(Z)
484 -- F(Z) = Z^2 + exp(sin(Z))*sin(Z)
485 -- F(Z) = Z^2 + (exp(cot(Z))/ Z) + C
486 -- F(Z) = Z * det|X A Y B| + i * det|A Y B X| + C
487 -- F(Z) = Z^2 + det|X A Y B| + C
488 -- F(Z) = Z^2 * C^2 * det|X A Y B| + C
489 -- F(Z) = Z^3 * C^3 * det|X A Y B| - C
490 -- F(Z) = 2XY - i(X^2 - Y^3) + C
491 -- F(Z) = 3*X^3*Y^2 - iZ^2 + C
492 -- F(Z) = ZC^3 - Z^2*C^2 - CZ^3 + C\n&);
493 -- F(Z) = X^2*Y*Z^2 - Y^2*Z*sin(X) + X^2*cos(Y) + C
494 -- F(Z) = Y^3*Z^3 - ZX^2 - C^2 * sin(Y) + C
495 -- F(Z) = X^2*Y^2*Z^2*C^2 + CXYZ + iZ^3
496 -- F(Z) = X^3*Y^3*Z^3 + iCXZ^2
497 -- F(Z) = X^2*Y^8*Z^5 + iXYZC^4
498 -- F(Z) = X^2*Y^3*Z^4 + XY^2*Z^3 + Z^2 + C
499 -- F(Z) = XYZ^2*cos(X) + X^2 - Y^2 + BCZ + C
500 -- F(Z) = Z^2 + ABXY + C
The next fifty-two equations
are exhibited in the
brand new 2006 &M& line of fractal imagery.
501 -- F(Z) = Z^C + SIN C
502 -- F(Z) = (SQRT Z + C)
503 -- F(Z) = (SIN(Z+1))^Z
* COS(1/Z) + C
504 -- F(Z) = SIN(Z * COS
Z + C) + C
505 -- F(Z) = COS(Z^2 / C)
+ SIN(C / Z^2) + C
506 -- F(Z) = Z^C + C^Z +
C^(Z^2) + Z^(C^2)
507 -- F(Z) = ((Z + SIN(Z)
+ C)^2 + C) / Z
508 -- F(Z) = SIN(COS(Z^2
+ C) + Z + C
509 -- F(Z) = SIN(Z^2) *
LOG(C^2) * (Z * C^(SQRT(Z)) + C
510 -- F(Z) = Z^2 + Z * C^(SQRT(Z))
511 -- F(Z) = SIN(Z^2 / C)
+ COS(Z / C) + Z^2 + C
512 -- F(Z) = ((Z^4 + C)
/ (Z^2 - C)) + Z^2 + C
513 -- F(Z) = ((Z^2 + C)^2
+ Z - C^3) / 2(Z^2 + C)
514 -- F(Z) = e^(Z^2) + e^2Z
- Z^2 - 1 + C
515 -- F(Z) = (2e^2Z / Z^3)
516 -- F(Z) = 3Z^2 * 3^(Z^3)
517 -- F(Z) = (1 + LOG(Z))^3
518 -- F(Z) = (Z + 1/Z)^2.5
519 -- F(Z) = (Z^(Z - 1)
+ 1)^2 + C
520 -- F(Z) = (1 / LOG(Z
521 -- F(Z) = C * (Z + 1/Z)
+ Z^(-.5) + C
522 -- F(Z) = Z^2 + Z^1.75
+ Z^1.5 + Z^1.25 + Z + C
523 -- F(Z) = ((Z^4 + Z^2
+ C)^2 / (Z^5 + Z^3 + Z + C))
524 -- F(Z) = (1 / SINH(1
/ Z^2)) + C
525 -- F(Z) = (Z ^ C + Z)
526 -- F(Z) = (Z + C^Z) /
527 -- F(Z) = (Z^C + C) /
528 -- F(Z) = (C^Z + C) /
529 -- F(Z) = Z^1.75 + C
530 -- F(Z) = (Z^2.5 / LOG(Z))
531 -- F(Z) = Z^2 * LOG(Z)
- C*Z + Z + C
532 -- F(Z) = SQRT(Z^6 +
Z + C) + C
533 -- F(Z) = (C * Z^2)^Y
534 -- F(Z) = SQRT(Z^4 +
Z^3) + Z + C
535 -- F(Z) = (((Z^2 + X
- 1)^2) / (2Z + X + 1)^2) + C
536 -- F(Z) = Z^2 + X + Y
- X^2 * Y^2 + C
537 -- F(Z) = Z^(Z^2 + C)
538 -- F(Z) = (Z + TAN(Z)
+ TAN(1/Z))^2 + C
539 -- F(Z) = Z^(C - 1) *
(1 - Z - Z^2) + C
540 -- F(Z) = LOG(1 / (COS(Z^2
541 -- F(Z) = Z^3 + (TAN(Z)
+ C * Z)^2
542 -- F(Z) = Z^2 * X - Z
543 -- F(Z) = Z^2 * CSC(Z^2)
544 -- F(Z) = (Z + LOG(Z))^3
545 -- F(Z) = Z^(LOG(Z^2))
546 -- F(Z) = (2Z^4 - Z +
C) / (Z + C)
547 -- F(Z) = Z^5C - Z^3C
548 -- F(Z) = Z^2 + ((Z^3)
/ (TAN(Z)^2))) + C
549 -- F(Z) = C^(Z^2 - Z
550 -- F(Z) = Z^7 + ((Z^5)
/ (5 - Z)) + ((Z^3) / (3 - Z)) + Z / C
551 -- F(Z) = (TAN(C * Z))^2
552 -- F(Z) = Z^3 + (1 +
LOG(Z))^2 + C
Lissajous Figures
A 3-D Lissajous figure is
created using three parametric equations, one each for the x, y, and z coordinates,
that is, each coordinate is a function of the independent parameter, time, t.
These equations are sinusoidal functions (sines and cosines) so they are periodic,
with the actual period depending on what values you enter. The values you input
in these functions are the coefficients a and b, and the exponents i, j, and
k. The value of t ranges from 0 to the number of spheres plotted minus one.
Thanks to Aaron C. Caba for the info.
I used five different sets of equations. Here they are:
x(t) = r * (sin(a*t) * (cos(b*t)^i))
y(t) = r * (sin(a*t) * (sin(b*t)^j))
z(t) = r * (cos(a*t)^k)
x(t) = r * (sin(a*t) * (cos(b*t)^i))
y(t) = r * (cos(a*t) * (cos(b*t)^k))
z(t) = r * (sin(a*t)^k)
x(t) = r * (sin(a*t) * (sin(b*t)^i))
y(t) = r * (sin(a*t) * (cos(b*t)^j))
z(t) = r * (sin(a*t)^k)
x(t) = r/4 * (a * sin(2*(t-pi/13))^i)
y(t) = r/4 * (-b * cos(t)^j)
z(t) = r * (sin(a*t)^k)
x(t) = r * (sin(a*t) * (cos(a*t)^i))
y(t) = r * (sin(b*t) * (sin(b*t)^j))
z(t) = r * (sin(t)^k)
Spherical Harmonics
Spherical harmonics are expressions
in three-dimensional spherical coordinates which are primarily used to describe
the theoretical hybrid electron orbital shapes in molecules. The three coordinates
are r (for radius), theta (degrees in the traditional x-y plane), and phi (degrees
in the y-z plane). You may also recognize this way of laying out spatial coordinates
from Star Trek's &210 mark 45& designation for navigation as the degrees
in theta and phi. As with the rectangular coordinates, x, y, and z, we can describe
any point in three dimensional space using such a coordinate system. All types
of scientists and engineers use spherical and cylindrical (rho, theta, and z)
coordinate systems in addition to the familiar rectangular system to analyze
various physical phenomena.
Here are a few of the examples
we have used to produce our mathematical &flying saucers:&
r = (cos (theta))^2 + (cos(2
* theta))^4 + sin(4 * phi)
r = (cos(12 * theta))^5 +
(cos(8 * theta))^3 + cos(6 * theta)
r = 2 * (cos(6 * theta))^6
- 4 * (cos(4 * theta))^4 - 2 * (cos(2 * theta))^2
rho = (sin(theta))^4 + (sin(2
* theta))^2 + e ^ (1 - sin(z))
rho = 4 * (cos(4 * theta))^4
- 2 * (cos(2 * theta))^2 + (1 + cos (z))^2
You can experiment with an
infinite number of possibilities. You will soon discover what
each coefficient, exponent, and function does to the overall shape
of the object. Happy Hunting!
large and extended
study of over 640 images, go .
Knots are a colloquial term
for three dimensional figures very much akin to Lissajous figures. The program
I used to generate these &knots& was created by Lloyd Burchill, a
most clever programmer and mathematician who created this shareware gem. His
program allows one to generate multiple parametrically calculated figures and
their interaction produces some very surprising results. Additionally there
are some special techniques and features available to produce other very tricky
designs. You can get it .
You can e-mail Mr. Burchill at .
Also, there's the very clever
and interesting KnotPlot software available from Robert Scharein .
I recommend this one, indeed.
I have yet to delve into its mysteries.
Polyhedra, the plural of polyhedron,
are three-dimensional solid figures with many geometrical faces
to them. There are five commonly known regular polyhedra, regular
meaning all faces are congruent and all angles are congruent.
Tetrahedron
equilateral triangle
Hexahedron
Octahedron
equilateral triangle
Dodecahedron
Icosahedron
equilateral triangle
There is information regarding
formulas to find the volumes, surface areas, inscribed radii, and circumscribed
radii of the above polyhedra HERE.
There are also the Archimedean
solids, solid shapes whose faces are all regular polygons of two
or more kinds, and whose vertices are all identical. There are
13 different kinds. Two (the snub cube and snub dodecahedron)
come in paired mirror-image forms. Eleven of these solids can
be formed by truncating (chopping the corners off) simpler solids.
They have pleasingly symmetrical crystalline shapes, and are described
below. These eleven are:
&Truncated Tetrahedron
&8 faces (4 triangles,
4 hexagons)
&Truncated Cube
&14 faces (8 triangles, 6 octagons)
&Truncated Octahedron
&14 faces (6 squares, 8 hexagons)
&Cuboctahedron
&14 faces (8 triangles, 6 squares)
&Truncated Dodecahedron
&32 faces (20 triangles, 12 dodecagons)
- soccer ball pattern
&Truncated Icosahedron
&32 faces (12 pentagons, 20 hexagons)
- soccer ball / fullerene shape
&Icosidodecahedron
&32 faces (20 triangles,
12 pentagons)
&Small Rhombicuboctahedron
&26 faces (8 triangles, 18 squares)
&Great Rhombicuboctahedron
&26 faces (12 squares, 8 hexagons,
6 octagons)
&Small Rhombicosidodecahedron
&62 faces (20 triangles, 30 squares,
12 pentagons)
&Great Rhombicosidodecahedron
&62 faces (30 squares, 20 hexagons,
12 dodecagons)
Thanks to Grant Hutchison for the info.
Stellated Polyhedra
Below are the stellated forms of the simple
Platonic solids we use in our scenes.
Stellated Octahedron
Stellated Dodecahedron
Stellated Icosahedron
The cube and tetrahedron have no stellated forms.
The stellations come in several different forms,
depending on the shape of their faces. The octahedron has a single stellation,
the dodecahedron has three, and the icosahedron has fifty-nine different stellations,
fifteen of which are given here.
The octahedron has only one stellation, the
Stella Octangula.
The three stellations of the dodecahedron are:
Small stellated dodecahedron
Great dodecahedron
Great stellated dodecahedron
There are fifty-nine stellations of the icosahedron.
A selection of the fifteen we use are:
Compound of five tetrahedra
Compound of ten tetrahedra
Compound of five octahedra
First stellation
Second stellation
Third stellation
Fourth stellation
Sixth stellation
Seventh stellation
Ninth stellation
Tenth stellation
Fourteenth stellation
Fifteenth stellation
Great icosahedron
Final stellation
Thanks to Grant Hutchison and Magnus Wenninger
for the info.
Non-Convex Uniform &Starface&
Uniform polyhedra are solids
whose faces are all regular polygons, and whose vertices are identical. The
convex uniform polyhedra (or Starfaces) are the five Platonic solids, the thirteen
Archimedean solids, and an infinite series of rather mundane prisms and antiprisms.
Non-convex uniforms have faces
which interpenetrate, making very complicated and pleasing solids.
Four of these are the small stellated dodecahedron, the great
stellated dodecahedron, the great dodecahedron and the great icosahedron.
There are at least another
53 forms, 21 of which are here. The 21 solids below can be grouped into seven
families. They have been numbered from one to seven arbitrarily. Each member
of a family shares some of its faces with other members.
The seven families are:
&Small Dodecahemicosahedron
&12 pentagrams, 10
&Great Dodecahemicosahedron
&12 pentagons, 10 hexagons
&Dodecadodecahedron
&12 pentagrams, 12
Ditrigonal Icosidodecahedron
&12 pentagrams, 20
Ditrigonal Icosidodecahedron
&12 pentagons, 20 triangles
&Ditrigonal
Dodecahedron
&12 pentagrams, 12
&Rhombidodecadodecahedron
&12 pentagrams, 12
pentagons, 30 squares
&Rhombicosahedron
&30 squares, 20 hexagons
&Icosidodecadodecahedron
&12 pentagrams, 12
pentagons, 20 hexagons
&Great Ditrigonal Dodecicosidodecahedron
&12 decagrams, 12 pentagons,
20 triangles
&Great Icosicosidodecahedron
&12 pentagons, 20 triangles,
20 hexagons
&Great Dodecicosahedron
&12 decagrams, 20 hexagons
&Small Ditrigonal Dodecicosidodecahedron
&12 pentagrams, 12
decagons, 20 triangles
&Small Icosicosidodecahedron
&12 pentagrams, 20
triangles, 20 hexagons
&Small Dodecicosahedron
&12 decagons, 20 hexagons
&Great Dodecicosidodecahedron
&12 pentagrams, 12
decagrams, 20 triangles
&Great Rhombidodecahedron
&12 decagrams, 30 squares
&Quasirhombicosidodecahedron
&12 pentagrams, 20
triangles, 30 squares
&Great Icosihemidodecahedron
&6 decagrams, 20 triangles
&Great Icosidodecahedron
&12 pentagrams, 20
&Great Dodecahemidodecahedron
&6 decagrams, 12 pentgrams
Families one and two are
similar - equalateral hexagons in one are replaced by pairs of triangles in
two. Four and five are also related - pentagrams replace pentagons, decagons
replace decagrams. Families six and seven have &small& versions that
aren't listed here - they are dissected versions of the rhombicosidodecahedron
and the icosidodecahedron, respectively, and they don't have any star-shaped
faces. Thanks to Grant Hutchison and Magnus Wenninger for the info.
For a scholarly look at Polyhedra
of all types, see George Hart's dynamite website .
Affine Transformations
(Due to the limitations of
web publishing, our notation of matrices, symbols, subscripts
etc. will be clumsily laid out...please bear with us...as our
tools improve, so will our presentation.)
As given by Barnsley in &,& an affine transformation is
a manipulation of a geometric set of points (here x1 and x2, or
just x) using matrices and column vectors such that:
w(x1,x2) = (ax1
+ bx2 + e , cx1 + dx2 + f)
A general affine two-dimensional
transformation, is given by:
where A is a 2 x 2 real matrix
and t is the column vector:
In graphic terms, the A matrix
transforms x by a linear transformation, which deforms
space relative to the origin (involving rotation and rescaling),
whereas the t vector merely translates (moves) the points once
the deformation is complete.
The matrix A can always be
written as:
where r1 and r2 are scaling
factors and g and h are rotation angles.
Barnsley continues in his
book to describe Iterated Function Systems, a way of describing objects created
by affine transformations. Using the letters a, b, c, d, e, and f as defined
above, he offers a typical a typical fern designation in tidier &IFS code:&
Notice he provides a number
p which corresponds to the probability that each of the four &w& transformations
will be used given each point (x1,x2) that is to be manipulated. All of the
p's must add up to one. Because of this probability factor, each time you generate
a spleenwort fern, it will be a slightly different one, just like Nature. Thus
we are not producing a &deterministic fractal,& as are Mandelbrot
and Julia sets (which are exactly reproducible), but more of a &random
iteration& fractal. See the Barnsley textbook for more info, illustrations,
IFS codes, etc.