如何在Maxima CAS中求解指数方程 [英] How can I solve exponential equation in Maxima CAS
问题描述
我在Maxima CAS中具有功能:
f(t):=(2 * exp(2 *％i *％pi * t)-exp(4 *％pi * t *％i))/4;
此处:
- t是0到1之间的实数
- 函数应在Mandelbrot集的主要心形心的边界上给出一个点
如何求解方程式:
eq1:c = f(t); (其中c是复数)
?
解决不起作用
solve(eq1，t); 结果为空列表
[] 该方程式的结果应给出复数点c的实数t(内角或旋转数)
感谢@JosehDoggie发表评论
我可以使用以下方法绘制初始方程:
加载(绘制)$f(t):=(2 * exp(％i * t)-exp(2 * t *％i))/4;draw2d(key ="main cardioid"，nticks = 200，参数(0.5 * cos(t)-0.25 * cos(2 * t)，0.5 * sin(t)-0.25 * sin(2 * t)，t，0,2 *％pi)，title ="M集的主心形")$ 或
draw2d(polar(abs(exp(t *％i)/2 -exp(2 * t *％i)/4)，t，0,2 *％pi)); 类似的图像(心形)在此处
Edit2:
(％i1)eq1:c = exp(％pi * t *％i)/2-exp(2 *％pi * t *％i)/4;％i％pi t 2％i％pi t％e％e(％o1)c = -----------------------2 4(％i2)解决(eq1，t);％i log(1-sqrt(1-4 c))％i log(sqrt(1-4 c)+1)(％o2)[t =--------------------------，t =--------------------------]％pi％pi 所以:
f1(c):= float(cabs(-％i * log(1-sqrt(1-4 * c))/％pi));f2(c):= float(cabs(-％i * log(1 + sqrt(1-4 * c))/％pi)); 但是效果不好.
也许我应该从头开始.我有:
- 复数c(=心形的边界)
- 实数t(从0到1，有时从0到2 * pi)
- 从t计算c的函数f:c = f(t)
我想找到从c计算t的函数:t = g(c)
测试值:
- t = 0，c = 1/4
- t = 1/2，c = -3/4
- t = 1/3，c = c = -0.125 + 0.649519052838329 *％i
- t = 2/5，c = -0.481762745781211 + 0.531656755220025 *％i
- t = 0.118033988749895 c = 0.346828007859920 + 0.088702386914555 *％i
- t = 0.618033988749895，c = -0.390540870218399 -0.586787907346969 *％i
- t = 0.718033988749895 c = 0.130349371041523 -0.587693986342220 *％i
load("to_poly_solve")$e:(2 * exp(2 *％i *％pi * t)-exp(4 *％pi * t *％i))/4-c $s:to_poly_solve(e，t)$s:maplist(lambda([e]，rhs(first(e)))，s)$/*解压缩％union的参数*/ratexpand(s); 输出
％i log(1-sqrt(1-4 c))％i log(sqrt(1-4 c)+1)(％o6)[％z7--------------------------，％z9--------------------------]2％pi 2％pi I have function in Maxima CAS :
f(t) := (2*exp(2*%i*%pi*t) - exp(4*%pi*t*%i))/4;
here:
- t is a real number between 0 and 1
- function should give a point on the boundary of main cardioid of Mandelbrot set
How can I solve equation :
eq1:c=f(t);
(where c is a complex number)
?
Solve doesn't work
solve( eq1,t);
result is empty list
[]
Result of this equation should give real number t ( internal angle or rotation number ) from complex point c
EDIT: Thx to comment by @JosehDoggie
I can draw initial equation using:
load(draw)$
f(t):=(2*exp(%i*t) - exp(2*t*%i))/4;
draw2d(
key="main cardioid",
nticks=200,
parametric( 0.5*cos(t) - 0.25*cos(2*t), 0.5*sin(t) - 0.25*sin(2*t), t,0,2*%pi),
title="main cardioid of M set "
)$
or
draw2d(polar(abs(exp(t*%i)/2 -exp(2*t*%i)/4),t,0,2*%pi));
Similar image ( cardioid) is here
Edit2:
(%i1) eq1:c = exp(%pi*t*%i)/2 - exp(2*%pi*t*%i)/4;
%i %pi t 2 %i %pi t
%e %e
(%o1) c = ---------- - ------------
2 4
(%i2) solve(eq1,t);
%i log(1 - sqrt(1 - 4 c)) %i log(sqrt(1 - 4 c) + 1)
(%o2) [t = - -------------------------, t = - -------------------------]
%pi %pi
So :
f1(c):=float(cabs( - %i* log(1 - sqrt(1 - 4* c))/%pi));
f2(c):=float(cabs( - %i* log(1 + sqrt(1 - 4* c))/%pi));
but the results are not good.
Edit 3 :
Maybe I shoud start from it. I have:
- complex numbers c ( = boundary of cardioid)
- real numbers t ( from 0 to 1 or sometimes from 0 to 2*pi )
- function f which computes c from t : c= f(t)
I want to find function which computes t from c: t = g(c)
testing values :
- t = 0 , c= 1/4
- t = 1/2 , c= -3/4
- t = 1/3 , c = c = -0.125 +0.649519052838329*%i
- t = 2/5 , c = -0.481762745781211 +0.531656755220025*%i
- t = 0.118033988749895 c = 0.346828007859920 +0.088702386914555*%i
- t = 0.618033988749895 , c = -0.390540870218399 -0.586787907346969*%i
- t = 0.718033988749895 c = 0.130349371041523 -0.587693986342220*%i
load("to_poly_solve") $
e: (2*exp(2*%i*%pi*t) - exp(4*%pi*t*%i))/4 - c $
s: to_poly_solve(e, t) $
s: maplist(lambda([e], rhs(first(e))), s) $ /* unpack arguments of %union */
ratexpand(s);
Outputs
%i log(1 - sqrt(1 - 4 c)) %i log(sqrt(1 - 4 c) + 1)
(%o6) [%z7 - -------------------------, %z9 - -------------------------]
2 %pi 2 %pi
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