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How To Draw Function Graphs (Mathematics / Analysis)

Empirical plotter for mathematical functions and calculus.

Due to overload issues, the plotter can no longer display extremely computational intensive functions. like the gamma functions and those which use this. Those can still be found in the source code .

With openPlaG mathematical function graphs can be drawn. Up to three graphs can be shown in one image. For this, input fields for three formulas are available. For the proper use of the plotter, it is advisable to activate JavaScript .

Draw plots the image of the graph, Reset sets all values back and Standard sets back display settings.

Syntax

The formulas f(x), g(x) and h(x) can contain the following symbols:

x Function variable x

0-9 Numbers, e.g. 123.45 Input values are hardly restricted (as long as you don't use numbers with more than 300 or so digits). The output value (result) at a non-log scale can only be up to 100000 (and -100000), this is also the maximum value for both axes.

Very large numbers can be written like 2.5E20 for 2.5*10 20. very small ones like 3E-10 for 3*10 -10. Decimal digits are exact up to a number of 12.

. Point or comma as decimal separator, e.g. 1.5 or 1,5

( ) [ ] < > < > Brackets, e.g. <[(1+x)/(2-x)+1]*3>/(2*x^2). are allowed in any amount. Each opened bracket must be closed again. What kind of bracket you use doesn't matter.

# as separator for formulas with multiple input values, e.g. scir(x#2)

asy Vertical asymptote for a given fixed value, e.g. asy(1) or asy(e)

Q Substitution for a self-defined formula.

- Basic arithmetic operations

+ Plus, e.g. x+1

- Minus, e.g. 1-x

* Times, this can only be omitted, when standing between a number and a letter. E.g. you can write 2x instead of 2*x. but not xsin(x) or ex.

/ : Divided by, e.g. 1/x or 1:x

Constants

e Euler's number: 2.718281828459

pi π. Pi: 3.1415926535898

go Relation of the golden ratio: 1.6180339887499

d Feigenbaum constant delta: 4.6692016091030

Functions

Nested functions like sin(pow(x#2/3)) or polynomials like 2*x^3-4*x^2+x+1 are no problem at all. Functions with multiple variables, like norm. can have the x at any one or at several positions. The standard position is shown in the examples.

- Basic functions

^ or pow Power, e.g. x^2 or pow(x#2) for x 2. Root can be written as e.g. x^(1/2) or x^.5 for square root of x, an exponential function like this: e^x for e x .

Roots of negative values can only be shown, if the numerator of the power is 1 and the denominator of the power is odd (e.g. x^(1/3) ). To calculate negative x-values for e.g. x^(2/3). you have to alter this function into (x^(1/3))^2

sqr Square root, e.g. sqr(x) as equivalent to x^(1/2).

exp Exponent, e.g. exp(x) as equivalent to e^x.

log Natural logarithm. e.g. log(x)

logn Logarithm to the base n, e.g. logn(2#x) for the binary (base 2) logarithm.

- Trigonometric and hyperbolic functions

sin Sine, sinus, e.g. sin(x)

cos Cosine, cosinus, e.g. cos(x)

tan Tangent, e.g. tan(x)

cot Cotangent, e.g. cot(x)

sin2 Sine square, e.g. sin2(x)

cos2 Cosine square, e.g. cos2(x)

tanh Hyperbolic Tangent, e.g. tanh(x)

coth Hyperbolic Cotangent, e.g. coth(x)

arsinh Area Hyperbolic Sine, e.g. arsinh(x)

arcosh Area Hyperbolic Cosine, e.g. arcosh(x)

artanh Area Hyperbolic Tangent, e.g. artanh(x)

arcoth Area Hyperbolic Cotangent, e.g. arcoth(x)

cat Catenary, e.g. cat(2#x) for 2*cosh(x/2). The first value is the constant a.

hubb Hubbert curve, e.g. hubb(x) for 1/(2+2*cosh(x)).

L Langevin function, e.g. L(x) for coth(x)-1/x.

deg Converts the radian number to the equivalent number in degrees, e.g. deg(pi)

- Non-differentiable functions

abs Absolute value, e.g. abs(x)

min Minimum of several values, e.g. min(1#x#x^(1/3)) as minimum of 1, x and third root of x.

max Maximum of several values, e.g. max(abs(x)#x*x) as maximum of the absolute value of x and x 2 .

% Modulo division, whole-numbered remainder, e.g. 10%x

fmod Modulo division, floating point remainder, e.g. fmod(x#1) displays only the position after the decimal point of the input value.

R Round, e.g. R(x#2) rounds two decimal places, R(x) rounds to an integer.

R0 Floor (rounding down), e.g. R0(x)

R1 Ceil (rounding up), e.g. R1(x)

dist Distance function, e.g. dist(x) gives the distance to the nearest integer.

prime Prime number function, e.g. prime(x) This returns the next lower prime number (or x itself, if prime) for all x≥2 and x≤100000. At all four prime functions, non-integers are rounded.

prime1 Prime number detecting function, e.g. prime1(x) Displays a number only if prime, else 0. To find all prime numbers in an interval, the span of the x-axis shouldn't be wider than the width of the image (usually 500) and you should switch off poles.

prime2 Distinct prime factor counting function, e.g. prime2(x) returns the amount of different prime factors for an integer.

prime3 Prime factor counting function, e.g. prime3(x) returns the amount of prime factors for an integer, including multiples. E.g. prime2(4) = 1, whereas prime3(4) = 2. If prime3(x) = 1, then x is prime.

div Divisor function, e.g. div(x) returns the number of divisors of an integer. Non-integers are rounded.

dig Digit sum, e.g. dig(x) returns the digital sum of an integer. Non-integers are rounded, - is ignored.

dig2 Iterated (one-digit) digit sum, e.g. dig2(x) returns the iterated digital sum of an integer.

adig Alternating digit sum, e.g. adig(x) Non-integers are rounded, - is ignored.

fac Factorial, e.g. fac(x) Non-integers are rounded.

H Heaviside step function, e.g. H(x) 0, if x≤0, else 1.

Hm Multivariate Heaviside step function, e.g. Hm(x*x-1#sin(x)) 0, if at least one value ≤0, else 1. Enter as many arguments as you want.

sig Signum function (sign function), e.g. sig(x)

haar Haar wavelet, e.g. haar(x)

gcf Greatest common factor (or greatest common divisor, gcd), e.g. gcf(8#x) returns the greatest common factor between two integers. Non-integers are rounded.

lcm Least common multiple, e.g. lcm(8#x) returns the least common multiple between two integers. Non-integers are rounded.

mo Möbius function, e.g. mo(x) returns for all positive integers 0, if divisible by a square>1, -1 if it has an odd number of distinct prime factors and 1 if it has an even number of distinct prime factors. Non-integers are rounded. Values are allowed up to 100000.

toti Euler's totient function, e.g. toti(x) counts all positive integers less than x that are comprime to x. Non-integers are rounded.

odd Find odd numbers, e.g. odd(x) returns numbers only when odd. Non-integers are rounded.

even Find even numbers, e.g. even(x) returns numbers only when even. Non-integers are rounded.

bin Binomial coefficient, e.g. bin(4#x) The two values are n and k. Non-integers are rounded.

tri Triangle curve, e.g. tri(1#2#x) The first value is the period, the second is the amplitude.

rect Rectangle curve, e.g. rect(1#-1#2#x) The first value is the upper limit, the second is the lower and the third is the period.

saw Sawtooth wave, e.g. saw(2#1#x) The first value is the period, the second is the amplitude.

saw2 Reverse sawtooth wave, e.g. saw2(2#1#x) The first value is the period, the second is the amplitude.

ramp Ramp function, e.g. ramp(1#2#1#x) The first value is the start value, the second is the end value and the third is the height.

ramp2 Reverse ramp function, e.g. ramp2(1#2#1#x) The first value is the start value, the second is the end value and the third is the height.

trap Trapezium (trapezoid) function, e.g. trap(-4#-1#3#2#3#x) The first value is the start value of the climb, the second is the end value of the climb, the third is the height, the fourth is the start value of the descent and the fifth is the end value of the descent.

poly Polygon or chart line, e.g. poly(-4#2#-3#4#-2#1#-1#0#0#3#1#2#2#-1#3#3#4#1#x) gives a chart, respectively a half polygon. Here, (-4,2) is connected to (-3,4), this to (-2,1) and so on. The first value of each pair is the x-value, the second one is the y-value. The x-values must increase with each step. To get a full polygon, enter a second term with the same start and end points, like poly(-4#2#0.5#-4#4#1#x)

rand Integer random number between two integers, e.g. rand(0#2) returns 0, 1 or 2 (Mersenne twister is used for generating).

rand2 Random number between two numbers with decimal places (maximal 9), e.g. rand2(0#1#3) returns a number with three decimal places between 0 and 1 (Mersenne twister, too).

- Probability functions and statistics

norm Normal or Gaussian distribution, e.g. norm(0#1#x) for the standard normal distribution. The first value is the expected value, the second is the standard deviation.

phi Φ. Cumulative Gaussian distribution function, e.g. phi(0#1#x) This is an approximation based on the displayed interval. It delivers reasonable values, if the normal distribution in the chosen interval starts at very low values near 0. A common display of both functions is advisable.

lnorm Log-normal distribution, e.g. lnorm(0#1#x) The first value is the mean, the second is the standard deviation.

cau Cauchy distribution or Lorentz distribution, e.g. cau(0#1#x) for the standard Cauchy distribution. The first value is the location parameter, the second is the scale parameter.

lapc Laplace distribution, e.g. lapc(0#1#x) The first value is the location parameter, the second is the scale parameter. The second

parameter must be >0.

logd Logistic distribution, e.g. logd(1#2#x) The first value is the location parameter, the second is the scale parameter.

hlogd Half-logistic distribution, e.g. hlogd(x)

rlng Erlang distribution, e.g. rlng(5#1#x) The first value is the shape parameter, the second is the rate parameter. The first parameter must be a natural number.

pon Exponential distribution, e.g. pon(1#x) The first value is the rate parameter.

cosd Raised cosine distribution, e.g. cosd(0#1#x) The first value is the location parameter, the second is the scale parameter. cosd is defined in the interval [location-scale;location+scale].

sechd Hyperbolic secant distribution, e.g. sechd(x)

kum Kumaraswamy distribution, e.g. kum(2#3#x) The first two values are the shape parameters a and b.

levy Lévy distribution, e.g. levy(1#x) The first value is the scale parameter.

rlgh Rayleigh distribution, e.g. rlgh(1#x) The first value is the scale parameter.

wb Weibull distribution, e.g. wb(2#1#x) The first value is the shape parameter, the second is the scale parameter.

wig Wigner semicircle distribution, e.g. wig(1#x) The first value gives the radius.

igauss Inverse Gaussian distribution, e.g. igauss(1#0.25#x) The first value is the shape parameter, the second is the scale parameter.

par Pareto distribution, e.g. par(2#1#x) The first value is the location parameter, the second is the shape parameter.

shg Shifted Gompertz distribution, e.g. shg(0.5#1#x) The first value is the scale parameter, the second is the shape parameter, both must be >0.

brw Relativistic Breit-Wigner distribution, e.g. brw(1#2#x) The first value is the mass of the resonance, the second is the resonance's width and the third is the energy.

gen Generalized extreme value distribution, e.g. gen(0#1#0.2#x) The first value is the location parameter, the second is the scale parameter and the third is the shape parameter.

Ft Fisher-Tippett distribution, e.g. Ft(1#2#x) The first value is the location parameter, the second is the scale parameter. The second parameter must be >0.

rossi Rossi distribution, or mixed extreme value distribution, e.g. rossi(0#3#1#4#x) The first four values are c1, c2, d1 and d2.

gum1 Gumbel distribution type 1, e.g. gum1(2#1#x) The first two values are the parameters a and b.

gum2 Gumbel distribution type 2, e.g. gum2(2#1#x) The first two values are the parameters a and b.

trid Triangular distribution, e.g. trid(1#2#4#x) The first value is the lower limit, the second is the most probable and the third is the upper limit.

- Discrete distributions

bind Binomial distribution, e.g. bind(5#0.4#x) The first value is the number of trials, the second is the success probability.

poi Poisson distribution, e.g. poi(3#x) The first value is λ. the second is the expected value.

skel Skellam distribution, e.g. skel(1#2#x) The first two values are the means of two different Poisson distributions.

gk Gauss-Kuzmin distribution, e.g. gk(x)

geo Geometric distribution (variant A), e.g. geo(0.8#x) The first value is a probability.

hgeo Hypergeometric distribution, e.g. hgeo(8#3#2#x) The first value is the total number of objects, the second is the total number of defective objects, the third is is the number of sample objects and the fourth the number of defective objects in the sample.

logs Logarithmic series distribution, e.g. logs(0.1#x) The first value is a probability.

zm Zipf-Mandelbrot law or Pareto-Zipf law, e.g. zm(100#1#2#x) The first three values are N, q and s. Maximum for N is 100.

uni Uniform distribution, e.g. uni(1#2#x) The first value is the lower limit, the second is the upper limit.

- Special functions

traj Trajectory parabola, path of a thrown object, e.g. traj(45#20#9.81#x) The first value is the angle, the second is the speed (e.g. in meters per second). The third value is the gravitational acceleration (e.g. in m/s²), the normal value on earth for this is g = 9.81 m/s². The axes scale in this example is meters. Air resistance is ignored.

pll Parallel operator, as used, among others, for the calculation of parallel circuits and resistances, e.g. pll(20#30#x) Enter as many arguments as you want.

M1 Arithmetic mean, e.g. M1(2#3#x) Enter as many arguments as you want.

M2 Geometric mean, e.g. M2(2#3#x) Enter as many arguments as you want, only positive values are allowed.

M3 Harmonic mean, e.g. M3(2#3#x) Enter as many arguments as you want, only positive values are allowed.

M4 Root mean square, e.g. M4(2#3#x) Enter as many arguments as you want.

M5 Median, e.g. M5(2#3#x) Enter as many arguments as you want.

scir Semicircle curve, e.g. scir(x#1) for a semicircle with the radius 1. The formula is sqr(r*r-x*x), r gives the radius.

ell Semielliptic curve, e.g. ell(2#1#x) for a semiellipse with the horizontal radius 2 and the vertical radius 1. The formula is sqr((1-x*x/(a*a))*b*b).

ell2 Semi-superellipse or semi-hyperellipse, e.g. ell2(2#3#4#x) for a semiellipse with the horizontal radius 2, the vertical radius 3 and n=4.

lmn Lemniscate of Bernoulli, e.g. lmn(1#x) This returns a half lemniscate. For the other half, use -lmn(1#x)

lmn2 Lemniscate of Gerono, e.g. lmn2(x) This returns a half lemniscate. For the other half, use -lmn2(x)

lmn3 Lemniscate of Booth, e.g. lmn3(1#x) This returns a half lemniscate. For the other half, use -lmn3(1#x)

pyth Pythagorean theorem, e.g. pyth(x#1) The formula is c=sqr(a*a+b*b).

thr Rule of three, e.g. thr(x#1#2) The formula for thr(a#b#c) is f(x)=b*c/a.

fib Fibonacci numbers, e.g. fib(x) or fib(x#1) If the second value is 1, a continuous graph is shown, else a discrete.

dc Exponential decay, e.g. dc(5#1#x) The first value is the initial quantity, the second is the decay constant.

erf Gaussian error function, e.g. erf(x) For the computation its Taylor series is used.

HY4 Hyper4, also known as tetration or super-exponentiation, e.g. HY4(x#3) for x to the power of (x to the power of x). Here the maximum value can be excessed very quickly!

lambda Lambda function, e.g. lambda(x#3) for x to the power of (x to the power of (3-1)).

sgm Sigmoid function, e.g. sgm(x) for 1/(1+e^(-x)).

gom Gompertz curve, e.g. gom(2#-5#-3#x) The first value is the upper asymptote, the second is the parameter b and the third is the growth rate. Second and third value must be negative.

stir Stirling's approximation for large factorials, e.g. stir(x) The formula is (2*pi*x)^(1/2)*(x/e)^x.

omega Lambert-W function or Omega function or product log (approximation), e.g. omega(x)

bump Bump function psi, ψ. e.g. bump(x) for exp(-1/(1-x*x)) between -1 and 1, else 0.

srp Serpentine curve, e.g. srp(2#1#x) The formula is a*a*x/(x*x+a*b). The first two values are a and b.

bsc Gaussian bell-shaped curve, e.g. bsc(1#x) The formula is exp(-a*a*x*x), the first value is the shape parameter a.

gbsc Generalized Gaussian bell-shaped curve, e.g. gbsc(1#2#-1#x) for 1*exp(2*x-1*x*x).

- Programmable functions

bool Characteristic boolean function, e.g. bool(1/x) Returns nothing, if the input value is not defined, 0, if 0, else 1.

bool0 Defined boolean function, e.g. bool0(x) Returns 0, if the input value is 0 or not defined, else 1.

bool1 Undefined boolean function, e.g. bool1(prime1(x)) Returns nothing, if the input value is 0 or not defined, else 1.

con Condition function, e.g. con(0#sin(x)#1) The first value is the lower limit, the third is the upper limit. If the second value is between these two, the result is 1, else 0.

rcon Reverse condition function, e.g. rcon(0#sin(x)#1) The first value is the lower limit, the third is the upper limit. If the second value is between these two, the result is 0, else 1.

wcon Weighted condition function, e.g. wcon(0#sin(x)#1) Only returns the second value, if this lies between the first and the third value.

rwcon Reverse weighted condition function, e.g. rwcon(0#sin(x)#1) Only returns the second value, if this doesn't lie between the first and the third value.

&& (and) can be simulated with the minimum function, e.g. min< con[0#sin(x)#1] # con[0#cos(x)#1] >

|| (or) can be simulated with the maximum function, e.g. max< con[0#sin(x)#1] # con[0#cos(x)#1] >

⊕ (xor) can be simulated with the maximum minus the minimum function, e.g.

- Iterations (iterative functions)

y Previous function value, e.g. for y(0)+0.01 is 0 the initial value for y, the next value is the last result of the input value x and so on.

y2 Pre-previous function value, e.g. y2(1)+0.001

step Number of the iteration steps done, divided by the parameter value, e.g. step(100) counts up to five (at 500 px width).

mean Iterated arithmetic mean, e.g. mean(sin(x)) gives the arithmetic mean of the y-values returned to the so far reached x-values.

man Mandelbrot function, e.g. man(0#-1.9) for y(0)*y(0)-1.9.

Attention: derivative and integral with the iteration don't lead to very reasonable results. As well a logarithmic scale won't work here.

- Fractals

rsf Random singular function (a kind of devil's staircase), e.g. rsf(0#2) for y(a)+0.008*rand(0#1)*rand(0#1)*(b-a), from a (first value) to b (second value) at 500px width. The first value is the start point on the y-axis, the second is the average end value.

wf Weierstrass function, e.g. wf(x#0.5#17#10) The second value is a parameter between 0 and 1, the third value is a positive, odd integer. The second multiplied with the third must be larger than 1+3/2*pi. The fourth value is the number of steps done. In theory this is infinite, but here the maximum is 100.

blanc Blancmange curve, e.g. blanc(x#10) The second value is the number of steps done, maximum is 1000.

tak Takagi-Landsberg curve, e.g. tak(x#0.7#10) The second value is a parameter, which should be between 0 and 1. The third is the number of steps done, maximum is 1000.

Differential and integral equations

Derivatives within a function are written like this:

Source: rechneronline.de
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