Removable singularity

[_vince]

what happens when you take a function f(x), multiply it by (x-1) then divide by (x-1)? Would it be infinity at x=1? 

g(x) = f(x)*(x-1)/(x-1).  what is g(1)?

Try it out with the mouse:

Code: QB64: [Select]

1. const sw = 800
2. const sh = 600
3.  
4. dim shared pi
5. pi = 4*atn(1)
6.  
7. screen _newimage(sw, sh, 32)
8.  
9. do
10.         do
11.                 mx = _mousex
12.                 my = _mousey
13.                 mbl = _mousebutton(1)
14.                 mbr = _mousebutton(2)
15.                 mw = mw + _mousewheel
16.         loop while _mouseinput
17.  
18.         for yy=0 to sh
19.         for xx=0 to sw
20.                         u = (xx - sw/2)*0.01
21.                         v = (sh/2 - yy)*0.01
22.  
23.  
24.                         cmul uu, vv, u - (mx - sw/2)*0.01, v - (sh/2 - my)*0.01, u + 0.707, v - 0.707
25.                         cdiv x, y, uu, vv, u - 1, v
26.  
27.  
28.                 m = sqr(x*x + y*y)
29.                 a = (pi + _atan2(y, x))/(2*pi)
30.  
31.                 pset (xx, yy), hrgb&(a, m)
32.  
33.         next
34.         next
35.  
36.         _display
37.         _limit 60
38.  
39. loop until _keyhit=27
40.  
41. system
42.  
43. function hrgb&(h, m)
44.         r =  0.5 - 0.5*sin(2*pi*h - pi/2)
45.     g = (0.5 + 0.5*sin(2*pi*h*1.5 - pi/2)) * -(h < 0.66)
46.     b = (0.5 + 0.5*sin(2*pi*h*1.5 + pi/2)) * -(h > 0.33)
47.  
48.         dim n as long
49.     n = 16
50.    
51.     mm = m*500 mod 500
52.         p = abs((h*n) - int(h*n))
53.  
54.         rr = 255*r - 0.15*mm - 35*p
55.         gg = 255*g - 0.15*mm - 35*p
56.         bb = 255*b - 0.15*mm - 35*p
57.  
58.         if rr < 0 then rr = 0
59.         if gg < 0 then gg = 0
60.         if bb < 0 then bb = 0
61.  
62.         hrgb& = _rgb(rr, gg, bb)
63. end function
64.  
65.  
66. 'u + iv = (x + iy)*(a + ib)
67. sub cmul(u, v, xx, yy, aa, bb)
68.         x = xx
69.         y = yy
70.         a = aa
71.         b = bb
72.  
73.     u = x*a - y*b
74.     v = x*b + y*a
75. end sub
76.  
77. 'u + iv = (x + iy)/(a + ib)
78. sub cdiv(u, v, xx, yy, aa, bb)
79.         x = xx
80.         y = yy
81.         a = aa
82.         b = bb
83.  
84.     d = a*a + b*b
85.     u = (x*a + y*b)/d
86.     v = (y*a - x*b)/d
87. end sub
88.  

a singularity and two zeros
 

singularity removed with the mouse
 

[_vince]

Same thing but for the mighty

sin(x + iy)/(x + iy - mousex - imousey)

Code: QB64: [Select]

1. const sw = 800
2. const sh = 600
3.  
4. dim shared pi
5. pi = 4*atn(1)
6.  
7. screen _newimage(sw, sh, 32)
8.  
9.  
10. zz = 0.02
11.  
12. do
13.         do
14.                 mx = _mousex
15.                 my = _mousey
16.                 mbl = _mousebutton(1)
17.                 mbr = _mousebutton(2)
18.                 mw = mw + _mousewheel
19.         loop while _mouseinput
20.  
21.         for yy=0 to sh
22.         for xx=0 to sw
23.                         u = (xx - sw/2)*zz
24.                         v = (sh/2 - yy)*zz
25.  
26.  
27.                         'cmul uu, vv, u - (mx - sw/2)*0.01, v - (sh/2 - my)*0.01, u + 0.707, v - 0.707
28.                         'cdiv x, y, uu, vv, u - 1, v
29.  
30.                         sinz uu, vv, u, v
31.                         cdiv x, y, uu, vv, u - (mx - sw/2)*zz, v - (sh/2 - my)*zz
32.  
33.                 m = sqr(x*x + y*y)
34.                 a = (pi + _atan2(y, x))/(2*pi)
35.  
36.                 'pset (xx, yy), hrgb&(a, m)
37.                 pset (xx, yy), hrgb&(a, log(5000*m + 1))
38.  
39.         next
40.         next
41.  
42.         _display
43.         _limit 60
44.  
45. loop until _keyhit=27
46.  
47. system
48.  
49. function hrgb&(h, m)
50.         r =  0.5 - 0.5*sin(2*pi*h - pi/2)
51.     g = (0.5 + 0.5*sin(2*pi*h*1.5 - pi/2)) * -(h < 0.66)
52.     b = (0.5 + 0.5*sin(2*pi*h*1.5 + pi/2)) * -(h > 0.33)
53.  
54.         dim n as long
55.     n = 16
56.    
57.     mm = m*500 mod 500
58.         p = abs((h*n) - int(h*n))
59.  
60.         rr = 255*r - 0.15*mm - 35*p
61.         gg = 255*g - 0.15*mm - 35*p
62.         bb = 255*b - 0.15*mm - 35*p
63.  
64.         if rr < 0 then rr = 0
65.         if gg < 0 then gg = 0
66.         if bb < 0 then bb = 0
67.  
68.         hrgb& = _rgb(rr, gg, bb)
69. end function
70.  
71. function cosh(x)
72.     cosh = 0.5*(exp(x) + exp(-x))
73. end function
74.  
75. function sinh(x)
76.     sinh = 0.5*(exp(x) - exp(-x))
77. end function
78.  
79. sub sinz(u, v, xx, yy)
80.         x = xx
81.         y = yy
82.         u = sin(x)*cosh(y)
83.         v = cos(x)*sinh(y)
84. end sub
85.  
86. sub cosz(u, v, xx, yy)
87.         x = xx
88.         y = yy
89.         u = cos(x)*cosh(y)
90.         v =-sin(x)*sinh(y)
91. end sub
92.  
93.  
94. 'u + iv = (x + iy)*(a + ib)
95. sub cmul(u, v, xx, yy, aa, bb)
96.         x = xx
97.         y = yy
98.         a = aa
99.         b = bb
100.  
101.     u = x*a - y*b
102.     v = x*b + y*a
103. end sub
104.  
105. 'u + iv = (x + iy)/(a + ib)
106. sub cdiv(u, v, xx, yy, aa, bb)
107.         x = xx
108.         y = yy
109.         a = aa
110.         b = bb
111.  
112.     d = a*a + b*b
113.     u = (x*a + y*b)/d
114.     v = (y*a - x*b)/d
115. end sub
116.  
117.  
118.  

[MasterGy]

I like what you do, you do a very interesting thing.

[_vince]

Another note, the plot has two zeros and one pole (or singularity).  Stick it on the pole and it cancels but stick it on the other zero and notice the colours encircling it.  It goes from RGB to RGBRGB as you've increased the order of the zero to say (x-1)^2, a sort of period halving.

If you remember that exp(i w t) = sin wt + i cos wt, then (exp(i w t))^2 = sin 2wt + i cos 2wt, a sort of frequency doubling