What’s this function?

[SLD]

OK this is the integral function of . Since this has no definite integral I numerically integrated from 0 to the number. I simply used the Eudoxus method which is not particularly accurate but I think it’s close enough to see what it looks like. I did it from 0 to 20 using a 0.1 iteration. Pretty easy to do in excel. You need to divide the x axis numbers by 10.

But what is this function - roughly?

 

[Shadowy Man]

Looks like a sine curve plus a line.

 

[SLD]

But the function can’t decrease.

 

[skepticalbip]

But the function can’t decrease.

It won't if the sinewave amplitude is limited and the slope of the line is sufficient, the sum of the two will not decrease.

 

[Shadowy Man]

But the function can’t decrease.

It won't if the sinewave amplitude is limited and the slope of the line is sufficient, the sum of the two will not decrease.

Fit a line to it then subtract that off and see how well a sine fits the residuals.

 

[steve_bank]

I don't have a copy anymore, it may be on line. Check the CRC math handbook table of integrals.

If the Excel trig functions you are using are in radians mode you will need to evaluate from 0 to 2pi.

dt = .01
length = 2pi/dt vary the numerator to change limits of integration. 45deg = pi/4
integral = 0
t = 0
for i = 1 to lrngth
---y(i) = f(t)
---integral = integral + [y(i) * dt]
---t = t + dt
end

With modern computer speeds the brute force approach to integration works, You can use Excel Basic.

 

[repoman]

I would be shocked if it could be solved in a closed form. d

e^sin(x) could maybe have a induction rule for its power series as would the other part under the radical. Then you would have to multiply these two infinite power series together and figure that out. Integrating is easy after that.

Not sure if it can be done or how many terms would be need to figure out the function for coefficients for each power term. Multiplying infinite series gives me a headache and blurred vision.


for large values of x it will be very nearly periodic.


I got that the numerical value of the integral from 0 to 200 was 316.259... so not sure about your scale.

 

[SLD]

I don't have a copy anymore, it may be on line. Check the CRC math handbook table of integrals.

If the Excel trig functions you are using are in radians mode you will need to evaluate from 0 to 2pi.

dt = .01
length = 2pi/dt vary the numerator to change limits of integration. 45deg = pi/4
integral = 0
t = 0
for i = 1 to lrngth
---y(i) = f(t)
---integral = integral + [y(i) * dt]
---t = t + dt
end

With modern computer speeds the brute force approach to integration works, You can use Excel Basic.

WolframAlpha couldn’t integrate it, but it can numerically analyze it. I think it can only be numerically analyzed using zero as a starting point. But the graph of the integrand looks familiar. So I thought maybe there could be a close fit.

I tried a version of a*sin(b*x)+c*x and you can fit it for a while but eventually it starts oscillating down. This function can never decrease as the derivative function never goes below zero.

 

[SLD]

But the function can’t decrease.

It won't if the sinewave amplitude is limited and the slope of the line is sufficient, the sum of the two will not decrease.

Wait! I think this is close!



This is

I wonder how close one could get.

 

[Shadowy Man]

I tried a version of a*sin(b*x)+c*x and you can fit it for a while but eventually it starts oscillating down. This function can never decrease as the derivative function never goes below zero.

Maybe just need an exp(-x/c) on front with a long time constant c to get the ring down fitting right.

 

[repoman]

Is there a program that will generate the exact analytical coefficients of the power series of e^(sin(x)) and the power series of (arctan (x^2))^1/2 ?

Will these two power series be simple meaning that for coeff400 * x^400 that coeff400 will not have to have ~400 calculations to find the value of this coefficient?

Then a program that gives the exact analytical coefficients of the power series of their product?

Ok partially answering this

https://www.wolframalpha.com/widgets/view.jsp?id=f021a5566d8509939615e02a20f267e3



looking hairy. Looks like WA is doing this:

https://math.stackexchange.com/questions/735287/need-help-to-solve-taylor-series-of-e-sin-x

and for the other part

https://www.wolframalpha.com/widgets/view.jsp?id=f021a5566d8509939615e02a20f267e3

 

[steve_bank]

I don't have a copy anymore, it may be on line. Check the CRC math handbook table of integrals.

If the Excel trig functions you are using are in radians mode you will need to evaluate from 0 to 2pi.

dt = .01
length = 2pi/dt vary the numerator to change limits of integration. 45deg = pi/4
integral = 0
t = 0
for i = 1 to lrngth
---y(i) = f(t)
---integral = integral + [y(i) * dt]
---t = t + dt
end

With modern computer speeds the brute force approach to integration works, You can use Excel Basic.

WolframAlpha couldn’t integrate it, but it can numerically analyze it. I think it can only be numerically analyzed using zero as a starting point. But the graph of the integrand looks familiar. So I thought maybe there could be a close fit.

I tried a version of a*sin(b*x)+c*x and you can fit it for a while but eventually it starts oscillating down. This function can never decrease as the derivative function never goes below zero.

Except for basic functions I always used numerical integration. Simple algorithm checks are integrating sin and e^x. Integral e^x = e^x. Area = e^_upper_limit - e^x_lower_limit. Integral sin = -cos.

To check the algorith you are using online integrate e^x from 0 to 2. It should be e^2 – e^0 ,7.39 – 1 = 6.39

I ran it on Scilab
for t = 0 to 2PI
y(t) = exp(sin(t)) * sqrt(atan(t))^2)

I'll post the plots.

Adaptive predictor corrector kinds of algorithms to improve speeds and accuracy can be unstable. Especially with inputs that are not well behaved.

I always used simple rectangular integration with small intervals.

I use Open Office not Excel. The math works, I am trying to figure out how to access spread sheet cells from Basic.

Try using Scilab, it is free and professional grade. Multiple integration functions to experiment with.

 

[steve_bank]

I came up with 8.12 approx as the area using rectangular integration About 6000 steps. A simple algorithm is to decrease step size until the relative change in the result falls within a bound.

As a quick check for the exponential part the points at 0, pi and 2pi are correct. Sin 0 , pi or 2pi is 0 and e^0 = 1.

For th trig term the left end point is atan(0). As x gets large atan goes to 1.57. Square root of 1.57 is 1.23. The curve looks right. Asymptotic at 1.2 approximately.

Rectangular integration.

for 1 to n
integral = integral + f(x) * dx
end

clear
dt = .001
_length = floor(2*%pi/dt)
t = 0
s = 0
for i = 1:_length
...y1(i) = %e^sin(t)
...y2(i) = sqrt(atan(t^2))
...y3(i) = sqrt(atan(t^2)) * %e^sin(t)
...s = s + y3(i) * dt // integral summation
...t = t + dt
end

disp(s)

a = scf(1)
clf(a)
plot(y1)

b = scf(2)
clf(b)
plot(y2)

c = scf(3)
clf(c)
plot(y3)

 

[steve_bank]

e^sin(x)

 

[steve_bank]

atan

 

[steve_bank]

product