# Feynman's Calculus

by murshid_islam
Tags: calculus, feynman
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P: 2,168
 Quote by quetzalcoatl9 not to nitpick, but shouldn't it be $$\frac{d}{dt}$$ ?
Yes it was a typo.
$$\frac{d}{dt}F(t)=F'(t)=-2F(t)$$
while
$$\frac{d}{dx}F(t)=0$$
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P: 1,593
 Quote by lurflurf Here is an example find $$\int_0^\infty\exp(-x^2-1/x^2) dx$$ let $$F(t)= \int_0^\infty\exp(-x^2-t^2/x^2) dx$$ now we want F(1) F'(t)=-2F(t) so
You know what, I just don't see that. Can someone help me? When I take the derivative I get:

$$\frac{d}{dt}\int_0^{\infty}\exp(-x^2-t^2/x^2) dx=-2t\int_0^{\infty}\frac{1}{x^2}e^{-(x^2+\frac{t^2}{x^2})}dx$$

and thus I don't see how the derivative is -2F(t).
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P: 2,168
 Quote by saltydog You know what, I just don't see that. Can someone help me? When I take the derivative I get: $$\frac{d}{dt}\int_0^{\infty}\exp(-x^2-t^2/x^2) dx=-2t\int_0^{\infty}\frac{1}{x^2}e^{-(x^2+\frac{t^2}{x^2})}dx$$ and thus I don't see how the derivative is -2F(t).
That step (F'(t)=-2F(t)) was a little sudden so upon request I added a few intermidiate steps a few posts up. You can see the equality more easily if a substitution like u=t/x is made.
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P: 1,593
 Quote by lurflurf First thing you have a sign error in your derivative +t^2/x^2 should have a minus sign. That step (F'(t)=-2F(t)) was a little sudden so upon request I added a few intermidiate steps a few posts up. You can see that equality more easily if a substitution like u=t/x is made.
Very good Lurflurf. I see that now. Also, I took the minus sign out of parenthesis so I think I was Ok with that.
 P: 240 For the change of variable I am using these functions: u = u(x) = t/x (t fixed) <´¨¨´¨¨´¨k¨´¨´´´´j´´¨¨¨¨z¨´s¨s¨s´s´s´´´<
 P: 240 Sorry for that post. Castilla.
P: 89
So if one integrates from -infinity to infinity, can you always change the range (or whatever it's called) of integration to 0 to infinity, and multiply the remaining integral by 2? Or is that a property of the sin that makes it symmetrical to where the negative side doesn't affect it? In other words, why doesn't a -2 come out?

 Quote by lurflurf If you mean using differentiation with respect to a parameter (ie under the integral sign it can be done line this. $$\int_{-\infty} ^\infty \frac{Sin (x)}{x} dx =2\int_{0} ^\infty \frac{Sin (x)}{x} dx$$
How does this exp(i z) work? Is it like i to the z power?

I may have seen some exp functions with three variables like exp (x y z) or so. Does that mean anything?

 Quote by lurflurf Take f=exp(i z)/z
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P: 1,391
 Quote by nanoWatt So if one integrates from -infinity to infinity, can you always change the range (or whatever it's called) of integration to 0 to infinity, and multiply the remaining integral by 2? Or is that a property of the sin that makes it symmetrical to where the negative side doesn't affect it? In other words, why doesn't a -2 come out?
The property you described depends on whether a function is even or odd. An even function is a function such that f(-x) = f(x), while an odd function is one such that g(-x) = -g(x).

So, for example, cosine is an even function, since cos(-x) = cos(x), while sine is an odd function, since sin(-x) = -sin(x). The exponential function is neither odd nor even, as e^(-x) does not equal either e^(x) (for any arbitrary value of x) or -e^(x).

Because an even function looks the same in the x > 0 half plane as it does in the x < 0 half plane, if you have symmetric limits about x = 0, then integrating from -L to L of an even function is just like integrating from 0 to L twice. This is only true for even functions. For odd functions, the contribution from the negative half plane will cancel out that from the positive half plane, so the result will be zero.

To summarize:

$$\int_{-L}^{L} dx f(x) = 2 \int_{0}^{L} dx f(x)~\mbox{if f(x) is even}$$
$$\int_{-L}^{L} dx f(x) = 0~\mbox{if f(x) is odd}$$

 How does this exp(i z) work? Is it like i to the z power? I may have seen some exp functions with three variables like exp (x y z) or so. Does that mean anything?

Have you ever heard of the imaginary unit i? It is the number defined such that $i^2 = -1$. With this number one can define the complex exponential function, which has the property that

$$e^{ix} = \cos x + i \sin x$$

Since exponentials typically have nicer properties than sines or cosines, by considering the integral of exp{ix}/x you might be able to get the integral for sin(x)/x in an easier fashion (and, you'll probably also get the integral of cos(x)/x out of it too, if that happens to be finite, which I don't think it is).
 P: 89 Thanks for the breakdown. So how would you expand exp (i z)? Is that the same as $$e^{iz}$$ ?
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 Quote by nanoWatt Thanks for the breakdown. So how would you expand exp (i z)? Is that the same as $$e^{iz}$$ ?
exp{z} is just another notation for e^z.

If z is a complex number z = x + iy (where x and y are real), then

e^{z} = e^{x}e^{iy} = e^{x}(cos(y) + i sin(y))

And e^{i z} = e^{-y + ix} = e^{-y}(cos(x) + i sin(x))
P: 1
 Quote by HallsofIvy Differentiating under the integral: Leibniz's rule- $$\frac{d}{dx}\int_{a(x)}^{b(x)} f(x,t)dt= \int_{a(x)}^{b(x)}\frac{\partial f(x,t)}{\partial x}dt+ \frac{da(x)}{dx}f(x,a(x))- \frac{db(x)}{dx}f(x,b(x))$$. ...
It seems to me that the signs on the last two terms of the right-hand side are reversed; should the formula be
$$\frac{d}{dx}\int_{a(x)}^{b(x)} f(x,t)dt= \left(\int_{a(x)}^{b(x)}\frac{\partial f(x,t)}{\partial x}dt\right)+ \frac{db(x)}{dx}f(x,b(x))- \frac{da(x)}{dx}f(x,a(x))$$
?

I guess so, but I haven't figured out how to prove this formula yet....
Emeritus