File:TrivFctRootExm.pdf
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[edit]DescriptionTrivFctRootExm.pdf |
English: The function g: R→R is a trivial 5th functional root of f: R+→R+ defined by f(x) = sin(x). The computation of f(π/6) = 1/2 = g5(π/6) is shown. See File:TrivFctRootDef.pdf for the definition of g, given an arbitrary f: R+→R+ and n∈N, and the computation of gn(x) = f(x) for all x∈R+. |
Date | |
Source | Own work |
Author | Jochen Burghardt |
Other versions | File:TrivFctRootExm.pdf * File:TrivFctRootExm svg.svg * File:TrivFctRootDef.pdf |
C source code for example coordinates |
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#include <stdio.h>
#include <math.h>
static int const n = 5;
static double f(double x) { return sin(x); }
static double g(double x) {
if (0.0 <= x ) return atan(x) - M_PI;
if (-(n-1)*M_PI/2 <= x && x < 0 ) return x - M_PI/2;
if (-n*M_PI/2 <= x && x < -(n-1)*M_PI/2) return f(tan(x+n*M_PI/2));
return 0.0;
}
int main(void) {
int i;
double gi;
while (1) {
printf("enter:\n");
scanf(" %lf",&gi);
for (i=0; i<=n; ++i) {
printf("g^%d(x) = %f\n",i,gi);
gi = g(gi);
}
}
return 0;
}
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Gnuplot source code |
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# given:
n = 5
f(x) = sin(x)
# trivial function root:
g(x) = ( 0 <= x ? atan(x) - pi \
: -(n-1)*pi/2 <= x && x < 0 ? x - pi/2 \
: -n*pi/2 <= x && x < -(n-1)*pi/2 ? f(tan(x+n*pi/2)) \
: 0 \
)
xmin = -8
xmax = 7
ymin = -8
ymax = 2
set xrange[xmin:xmax]
set yrange[ymin:ymax]
set grid
set xtics pi/2
set ytics pi/2
set format x "%5.2f"
set format y "%5.2f"
set key right bottom
# coordinate cross:
set arrow from xmin , 0 to xmax , 0 nohead lt 8
set arrow from 0 , ymin to 0 , ymax nohead lt 8
# shift ranges:
set arrow from -1*pi/2 , ymin to -1*pi/2 , ymax nohead lt 5
set arrow from -2*pi/2 , ymin to -2*pi/2 , ymax nohead lt 5
set arrow from -3*pi/2 , ymin to -3*pi/2 , ymax nohead lt 5
set arrow from -4*pi/2 , ymin to -4*pi/2 , ymax nohead lt 5
set arrow from -5*pi/2 , ymin to -5*pi/2 , ymax nohead lc 5
# 1 input/output example f:
set arrow from 0.523599 , 0 to 0.523599 , 0.500000 lt 1
# 5 input/output examples g:
set arrow from 0.523599 , 0 to 0.523599 , -2.659245 lt 2
set arrow from 0.523599 , -2.659245 to -2.659245 , -2.659245 lt 2
set arrow from -2.659245 , -2.659245 to -2.659245 , -4.230041 lt 2
set arrow from -2.659245 , -4.230041 to -4.230041 , -4.230041 lt 2
set arrow from -4.230041 , -4.230041 to -4.230041 , -5.800837 lt 2
set arrow from -4.230041 , -5.800837 to -5.800837 , -5.800837 lt 2
set arrow from -5.800837 , -5.800837 to -5.800837 , -7.371634 lt 2
set arrow from -5.800837 , -7.371634 to -7.371634 , -7.371634 lt 2
set arrow from -7.371634 , -7.371634 to -7.371634 , 0.500000 lt 2
plot x with lines lt 7
replot [x=0:xmax] f(x) with lines lw 3 lt 1 title"f(x)"
replot g(x) with lines lw 3 lt 2
pause -1
|
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- You are free:
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- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
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File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 09:14, 6 August 2020 | ![]() | 3,087 × 2,045 (11 KB) | Jochen Burghardt (talk | contribs) | exchange role for f and g to fit with article naming |
08:58, 6 August 2020 | No thumbnail | 0 × 0 (11 KB) | Jochen Burghardt (talk | contribs) | show g: R+ --> R rather than g: R --> R | |
07:40, 6 August 2020 | No thumbnail | 0 × 0 (11 KB) | Jochen Burghardt (talk | contribs) | Uploaded own work with UploadWizard |
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