Fractal
Formulas
Researched
Researched
|
Fractal
Formulas Researched |
For me, the single most interesting aspect
of doing
mathematical fractal art is to create and explore new
complex algebraic expressions which generate imagery
never before seen by human eyes. I cannot tell you how many
nights I spent calculating and devising and generating new
expressions and imagery...so much sleep was lost (however
voluntarily) since I started in 1988...
Nearly all of the people doing fractal art
now, and there are
thousands, use canned software which employ someone else's
algorithms...and that is fine, for a far fewer number of people
have the appreciation for art as well as the technical know-how
to do their own math...
The explorer in me pushes me onward...listed
here are the 500
fractal expressions, functions of a complex number z and a
complex constant c, that were used in all of my work prior to 2004,
plus fifty-two new equations I recently devised this past year.
Feel free to use and re-use them to your heart's (and mind's) content.
(Note: The expressions ln, LN, and LOG all
refer to the natural logarithm
and sqrt and SQRT all refer to taking the square root, of course.
Furthermore, X is the real part of Z, Y is the imaginary part of Z, A is
the real part of C, and B is the imaginary part of C)
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 + C
176 -- F(Z) = Z^12cosX - Z^11sinY - Z^10tanY + C
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 - Z)) + C
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) + C
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) = Y^2 - B
254 -- F(X) = X^2 + atan(Y/X) + A; F(Y) = Y^2 - A
255 -- F(X) = 1 - X - Y^2 + A; F(Y) = 1 - Y + X^2 + B
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) / X]
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 X + C
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 + XY + C
278 -- F(Z) = (Z ^ 2 + 1) ^ 2 / (Z + C) ^ 2
279 -- F(Z) = (Z ^ 2 + C + 1) ^ 2 / (Z - C - 1) ^ 2
280 -- F(Z) = Z ^ 4 * sin Y + Z ^ 2 * cos X + XY + C
281 -- F(Z) = Z * cos (XY) + C
282 -- F(Z) = Z ^ 2 * cos (X ^ 2 + Y ^ 2) + C
283 -- F(Z) = Z ^ (2 + ln C)
284 -- F(Z) = Z ^ (9/7) + C
285 -- F(Z) = Z ^ 5 * (1 - Z - (Z + C) ^ 2) + C
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 + cos(Z)
315 -- F(Z) = (Z ^ 2 * ln Z + Z + C) / (sin(Z)) ^ 2
316 -- F(Z) = Z ^ 4 + (cos X) ^ 2 + (sin Y) ^ 2 + C
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) + C
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 + C)
324 -- F(Z) = sqrt(Z ^ 6 - Z ^ 3 + C)
325 -- F(Z) = sqrt(ln (Z ^ 2) + Z ^ 2 * ln Z + C)
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) + C
337 -- F(Z) = Z ^ 2 * cos X * cos Y * exp(-Y) + C
338 -- F(Z) = XYZ * sin X * sin Y * exp(Z) + C
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 + C
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' + C
347 -- F(Z) = Z ^ 5 - C * Z ^ 3 - C' * Z ^ 2 + Z' + C
348 -- F(Z) = Z ^ 4 * Z' ^ 2 - Z ^ 3 * Z' + C
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 / Z ^ 2 + C
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 + C
358 -- F(Z) = XY^2Z^3 - X^3Y^2Z + X^2Y^2Z^2 + C
359 -- F(Z) = Z ^ 4 - X ^ 2 * cos(Y) + Y * sin(X) + C
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 - 1)
364 -- F(Z) = (Z ^ 2 - Z ^ 3 + 1) / (Z ^ 4 + C)
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 + C
374 -- F(Z) = XY^2Z + X^2YZ^4 + C
375 -- F(Z) = Z^2*sqrt(XY) + XY^2Z^4*sqrt(XY) + C
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 + A
393 -- F(Z) = Z^5 + Im(Z^4 + Z^3 + Z^2) + CZRe(Z^2 + C) + C
394 -- F(Z) = Z^3 + Z^2*cosY + ZsinX + C
395 -- F(Z) = Z^4 + (Z^2 / sinY) + (CZ^3 / cosX) + C
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)
+ C
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) / 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)) + C
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) + C
516 -- F(Z) = 3Z^2 * 3^(Z^3) - 3^Z + C
517 -- F(Z) = (1 + LOG(Z))^3 + C
518 -- F(Z) = (Z + 1/Z)^2.5 + C
519 -- F(Z) = (Z^(Z - 1) + 1)^2 + C
520 -- F(Z) = (1 / LOG(Z + C))
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) / C
526 -- F(Z) = (Z + C^Z) / C
527 -- F(Z) = (Z^C + C) / Z
528 -- F(Z) = (C^Z + C) / Z
529 -- F(Z) = Z^1.75 + C
530 -- F(Z) = (Z^2.5 / LOG(Z)) + C
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 + C
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) - Z + 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 + C)))
541 -- F(Z) = Z^3 + (TAN(Z) + C * Z)^2
542 -- F(Z) = Z^2 * X - Z * Y + C
543 -- F(Z) = Z^2 * CSC(Z^2) + C
544 -- F(Z) = (Z + LOG(Z))^3 + C
545 -- F(Z) = Z^(LOG(Z^2)) - Z + C
546 -- F(Z) = (2Z^4 - Z + C) / (Z + C)
547 -- F(Z) = Z^5C - Z^3C + Z^C + C
548 -- F(Z) = Z^2 + ((Z^3) / (TAN(Z)^2))) + C
549 -- F(Z) = C^(Z^2 - Z + C)
550 -- F(Z) = Z^7 + ((Z^5) / (5 - Z)) + ((Z^3) / (3 - Z)) + Z / C
551 -- F(Z) = (TAN(C * Z))^2 + C
552 -- F(Z) = Z^3 + (1 + LOG(Z))^2 + C