Let's discuss Mathematics

[ParadigmShifter]

Arithmetic is pretty easy, let z = a+ib, w = c+id

z+w = (a+c) + i(b+d)

z.w = (a+ib)(c+id) = ac + iad + ibc + i²bd = ac + i(ad + bc) + (-1)bd = (ac-bd) + i(ad+bc)

For division, define conjugate of z = a+ib as z* = a-ib

Then z.z* = a² - iab + iab - i²b²
= a² - (-1)b² = a² + b²

which is |z|²

so z.z* = |z|² => z.z*/|z|² = 1 => z^-1 = z*/|z|² (EDIT: Note, |z| is real so division is well defined)

To divide, w/z = w.z^-1

Euler's formula comes from the power series expansion for sin and cos, multiply the series form of sin(x) by i and add it to the series for cos(x), you get the series for exp(ix)

I don't know about the Laplace transform, sorry.

 

[Atticus]

When doing complex arithmetic, you can just proceed like i was a variable, and at the end replace -1 for all i^2 (or powers bigger than that accordingly, so i^4 =(-1)^2=1 or i^6=(-1)^3 = -1).

If you want more intuitive understanding of it, the addition is like vector addition in R^2. The multiplication of two complex numbers a and b is a vector whose length is |a||b|, and angle from the real axis sum of angles of a and b.

what u(t) stands for in the situation that the Laplace transform of F(s) = 1/s is e^t times u(t).

Can you post the whole definition? Is u(t) perhaps the function that is transformed?

 

[IdiotsOpposite]

u(t) is the unit step function, of course. And on a related note...

How do you find the Laplace transform of a product of unit step functions?

 

[Japanrocks12]

Thanks, Paradigm and Atticus, that was very helpful.

Yeah, u(t) is the step function, which I also think I understand now. Also, I made an error in my initial post, it should read "inverse Laplace transform"

 

[Souron]

Complex addition and subtraction can easily be done on Cartesian coordinates (a+bi). For Complex multiplication and division, convert to polar notation(r<&#966, and then it's easy. (r2<&#966*(r2<&#966;2)=r1*r2<&#966;1+&#966;2. (r2<&#966/(r2<&#966;2)=r1/r2<&#966;1-&#966;2. Where < is the angle sign.

 

[Atticus]

u(t) is the unit step function, of course.

So it's 0 for t<0 and 1 for t>1? (I've encountered this by the name Heaviside function).

How do you find the Laplace transform of a product of unit step functions?

If it's indeed the aforementioned, why product? Aren't they all equal to the unit step function itself?

And just to make sure, this is the definition of Laplace transform we're talking about here:

Right?

 

[IdiotsOpposite]

So it's 0 for t<0 and 1 for t>0? (I've encountered this by the name Heaviside function).



If it's indeed the aforementioned, why product? Aren't they all equal to the unit step function itself?

And just to make sure, this is the definition of Laplace transform we're talking about here:

Right?

FIFY, and yes, it is the Heaviside function.

Indeed it is, but the product I'm looking to transform is u(t)*u(10-t). And I'm not sure how to take a product like that.

 

[ParadigmShifter]

Split it into separate cases.

 

[Atticus]

Yeah, what ParadigmShifter said. Draw a picture of both functions, and figure out what their product might be. You don't usually do this kinds of products algebraically.

 

[dutchfire]

Got a problem? Draw a picture!

The #1 rule for solving your science problems.

 

[Spoonwood]

Define a non-horizontal, non-vertical line as its corresponding algebraic equation. Describe the addition of two such lines, the difference of two such lines, the product of two such lines, and supposing that no point of the line happens at x=0, what you get when you divide one such line by another such line.

 

[Earthling]

Indeed it is, but the product I'm looking to transform is u(t)*u(10-t). And I'm not sure how to take a product like that.

duality of convolution and multiplication. Sorry about not being able to type up an equation easily, but you're looking for:


[/URL]

Step functions are quite easily obviously to see how that plays out.

But calling it the Heaviside function *cringes*

edit - really you're looking at the Laplace transform of a boxcar here so I googled that but didn't find a more convenient/easy trick to it, DTFT of course jogged my memory but I didn't feel like messing around more. Your product is obvious anyway before the transform and I'm sure it simplifies into something nice enough.

though you could go with the normalized boxcar/rectangular pulse and scale it and shift it then I guess, check your answer. (2/w sin(w/2)) I believe for the Fourier transform, guess I could work that out to s, I dunno, being lazy again here. Hope there was some help.

edit again - cancel any stupidity, just check wiki link which I somehow didn't find:

http://en.wikipedia.org/wiki/Rectangular_function

oh and actually splitting this into two step functions subtracted rather than multiplied is simpler yet, yeah, duh.

So [1-e^(-10s)]/ s ? Sorry to leave you hanging if that's not right

 

[Mise]

It was named after a guy named Heaviside IIRC. Although the fact that it has a heavy side and an empty side is a nice easy way of remembering it...

p.s. the rect function causes much hilarity when lecturers use "E" (or even "A") as the function that's being multiplied by the rect function.

 

[Atticus]

Yes, I think he was Oliver Heaviside, British guy who was a forerunner of distribution theory, although not rigorous enough with them.

One another funny name is John. A set can be John, if it satisfies certain conditions that I don't now remember. The name comes from mathematician Fritz John. I heard that someone published a paper entitled "Julia is John", about the fractal set Julia, and another guy published "Julia is broad".

 

[IdiotsOpposite]

Yes, Earthling, that's right. Thanks!

Now, there is one other question I dreamed up last night while I was asleep. The floor function is a discrete function. However, is the statement in the attachment true? Can the limit of a discrete function yield a continuous function?

 

[ParadigmShifter]

Can you not use the sandwich theorem to show that?

EDIT: floor(nx) <= nx
and floor(nx) > n(x-1) surely

EDIT2: Maybe have to switch those 2 around if x<0 or something Not started lubricating my maths brain with alcohol yet.

EDIT2: Had a sip of beer. Those would give different upper and lower bounds.

You probably need to show the error term -> 0 as n->+inf

EDIT4: Not enough beer obviously, since EDIT3 should follow EDIT2

EDIT5: Wolfram alpha doesn't evaluate the limit though...

 

[ParadigmShifter]

Define a non-horizontal, non-vertical line as its corresponding algebraic equation. Describe the addition of two such lines, the difference of two such lines, the product of two such lines, and supposing that no point of the line happens at x=0, what you get when you divide one such line by another such line.

Oh, BTW, the answer is a line, a line, a parabola, and a rational function that I can't be bothered to do the calculus on to be able to draw a graph Might do it later after a couple of beers.

 

[Mise]

couldn't you just say:

nx = floor(nx) + remainder
nx/n = floor(nx)/n + remainder/n

so when n >> remainder (i.e. n >> 1), remainder/n -> 0, and floor(nx)/n -> x

 

[ParadigmShifter]

Yeah that's what I was hinting at in my second EDIT2.

 

[IdiotsOpposite]

Actually, you might be onto something there. There is a function, frac(nx) which measures the fractional part of nx. For all x and n that I care about...

frac(nx)+floor(nx)=nx

So subtract frac(nx) from both sides and you get this:

floor(nx) = nx - frac(nx)

Then divide by n to get:

floor(nx)/n = x - frac(nx)/n

Take the limit as n goes to infinity, x doesn't change but frac(nx)/n is the important part. The numerator can never be >1, while the denominator grows without bound. Therefore, frac(nx)/n goes to 0 as n goes to infinity. Therefore, the limit of floor(nx)/n as n goes to infinity is x-0, or x.