Help me understand Scale Law in Integral

Scale Law in Integral

Suppose ∫ f(x)dx = F(x)

then ∫ f(c•x)dx = F(c•x)/c

Anybody knows what if c is a complex number, will the scale law still hold?

Re: Scale Law in Integral

hi,

as of such, there are no rules that say the contant shouldnt be a complex number. Complex numbers only convey more information abt a signal( for that matter) which has in addition to the magnitude in ordinary numbers, the phase content of the signal. And u would know it can again be represented in the magnitude phase form also. Moreover, sqrt(-1) also is a number, with additionaal interpretations, would explain this. It would absolutely satisfy the theorem! below, i have attached the Mathematica code.. and its o/p for u to verify

hope this helps, and clarifies ur doubt!

/cedance

Scale Law in Integral

Am,

I am getting lost in your explanation. My puzzle is just a pure mathematical integral question. It isn't related with the understanding of complex number or imaginary part of it.

I can't use Mathematica to verify every single function to prove or disprove the law.

I don't have clear memory about this question, which must be clarified by my college calculus instructor. But I do remeber there's certain codition for this scale law holding. It must be related with the poles of function in the range of the integral. Anybody has a clue of it?

Re: Scale Law in Integral

dazhen said:

Am,

I am getting lost in your explanation. My puzzle is just a pure mathematical integral question. It isn't related with the understanding of complex number or imaginary part of it.

I can't use Mathematica to verify every single function to prove or disprove the law.

I don't have clear memory about this question, which must be clarified by my college calculus instructor. But I do remeber there's certain codition for this scale law holding. It must be related with the poles of function in the range of the integral. Anybody has a clue of it?

Click to expand...

though i dont find anything different in ur reply, i hoped the above explanation was alright! anyways, to conclude whatever i have in mind, and trying to convince u,

it doesnt matter, whatever bla bla bla is inside the scale law, if it isnt a function of x and iff it isnt a fun of x, then the law holds good.. be it complex, sinusioidal, hyberbolic or what ever it is... u could have thought of a simple example... just... cos(sin a * x) here sin a could also be written as difference of complex exponentials...

when u integrate this... what would be the result? i think it s obvious, the result holds true... i am not in a position to explain more.. with ur question. may be if u could reply with ur doubts more precisely(for me), i would be in a position to.. but i see no rule or point in having a different law incase of complex exponentials!

btw, the example given, i dint ask u to input every example in mathematica... i wanted to makle u understand more clearer. that s it! u should know, i wouldnt ask u to input any function in mathematica or so!

/cedance

Re: Scale Law in Integral

Am, Ok. What you said is the scale law holds in any circumstance, no matter c is a real constant number or complex constant number. (If I don't misunderstand your point). Nevertheless, my colleagues disagree that. They thought it must be some limitations when expanding to the complex region. As I said, the poles of complex function are required for extra treatment.

I give out a simple contradictory example here,

∫cos(x)dx=sin(x)

Is this right: ∫cos(i•x)dx = sin(i•x)/i ???? i=√-1

The answer is not.


Eular Indentity tell us: cos(i•x) = [exp(x)+exp(-x)]/2, sin(i•x) = [exp(-x)-exp(x)]/2

∫cos(i•x)dx = ∫ {[exp(x)+exp(-x)]/2}dx = [exp(x)-exp(-x)]/2 = -sin(i•x) ≠ sin(i•x)/i

Mathematica, Matlab, etc are good tools for scientific or engineering numerical calculation. But for some fundamental stuffs, they're always screwed up. I wouldn't rely on computer for basic math problem.

Re: Scale Law in Integral

dear dazhen,

wad u hav mentioned is quite interesting,,,,,

however, i hav some doubt....how do u prove this one


i only know the cosθ = [exp(jθ)+exp(-jθ)]/2, sin(θ) = [exp(jθ)-exp(-jθ)]/2j

could u prove it to me,,,from Eular Indentity??....

thanxs....

Re: Scale Law in Integral

dazhen said:

Am, Ok. What you said is the scale law holds in any circumstance, no matter c is a real constant number or complex constant number. (If I don't misunderstand your point).

Click to expand...

You got it right

dazhen said:

Nevertheless, my colleagues disagree that. They thought it must be some limitations when expanding to the complex region. As I said, the poles of complex function are required for extra treatment.

Click to expand...

poles has nothing to do hre.. till its a constant. remember its constant and not a variable!

dazhen said:

I give out a simple contradictory example here,

∫cos(x)dx=sin(x)

Is this right: ∫cos(i•x)dx = sin(i•x)/i ???? i=√-1

The answer is not.

Click to expand...

all i can say is ur formula is wrong.

cos (x) = exp(j*x) + exp(-j*x) [Am leaving that constant here for convenience]
cos(hx) = exp(x) + exp(-x)

dazhen said:

Eular Indentity tell us: cos(i•x) = [exp(x)+exp(-x)]/2, sin(i•x) = [exp(-x)-exp(x)]/2

∫cos(i•x)dx = ∫ {[exp(x)+exp(-x)]/2}dx = [exp(x)-exp(-x)]/2 = -sin(i•x) ≠ sin(i•x)/i

Click to expand...

for ur question:

cos(ix) = exp(i*i*x) + exp(-i*i*x) = exp(-x) + exp(x) = cos (hx)

=> ∫ cos(hx) = sin(hx) = exp(x) - exp(-x) = exp(-i*ix) - exp(i*ix)
=> expression = -(sin(hx)) = sin(ix)/i
which is true!

dazhen said:

Mathematica, Matlab, etc are good tools for scientific or engineering numerical calculation. But for some fundamental stuffs, they're always screwed up. I wouldn't rely on computer for basic math problem.

Click to expand...

i understand ur confidence in u, but dont underestimate these programs! moreover, get ur basics right. it would solve the confusion. there s no disadvantage or degradation in working with a computer program for solving math. if u would know how to solve.. its a real headache banging ur heads on it.. this time i have typed it here.. u could see... its the most difficult thing... to type math... i could have used mathematica.. but to convince u! hope uget it right! am actually a bit tired after typin this! carry on mate!

/cedance

Re: Scale Law in Integral

arunmit168 said:

for ur question:

cos(ix) = exp(i*i*x) + exp(-i*i*x) = exp(-x) + exp(x) = cos (hx)

=> ∫ cos(hx) = sin(hx) = exp(x) - exp(-x) = exp(-i*ix) - exp(i*ix)
=> expression = -(sin(hx)) = sin(ix)/i
which is true!

Click to expand...

arunmit168, what is cos(hx)? hyperbolic function? what is expression? what's the difference, the integral and the expression? one is the other's negative. i don't know your meaning.

try this..............

dazhen said:

Eular Indentity tell us: cos(i•x) = [exp(x)+exp(-x)]/2, sin(i•x) = [exp(-x)-exp(x)]/2

∫cos(i•x)dx = ∫ {[exp(x)+exp(-x)]/2}dx = [exp(x)-exp(-x)]/2 = -sin(i•x) ≠ sin(i•x)/i

Click to expand...

dazhen, second one is not right. sin(i*x)=[exp(i*i*x)-exp(-i*i*x)]/2i. that's why you get the wrong conclusion.

Re: Scale Law in Integral

THz said:

arunmit168 said:

for ur question:

cos(ix) = exp(i*i*x) + exp(-i*i*x) = exp(-x) + exp(x) = cos (hx)

=> ∫ cos(hx) = sin(hx) = exp(x) - exp(-x) = exp(-i*ix) - exp(i*ix)
=> expression = -(sin(hx)) = sin(ix)/i
which is true!

Click to expand...

arunmit168, what is cos(hx)? hyperbolic function? what is expression? what's the difference, the integral and the expression? one is the other's negative. i don't know your meaning.

try this..............

dazhen said:

Eular Indentity tell us: cos(i•x) = [exp(x)+exp(-x)]/2, sin(i•x) = [exp(-x)-exp(x)]/2

∫cos(i•x)dx = ∫ {[exp(x)+exp(-x)]/2}dx = [exp(x)-exp(-x)]/2 = -sin(i•x) ≠ sin(i•x)/i

Click to expand...

dazhen, second one is not right. sin(i*x)=[exp(i*i*x)-exp(-i*i*x)]/2i. that's why you get the wrong conclusion.

Click to expand...

I think you're certainly right. I'm wrong at that point.

Re: Scale Law in Integral

THz said:

for ur question:
cos(ix) = exp(i*i*x) + exp(-i*i*x) = exp(-x) + exp(x) = cos (hx)
=> ∫ cos(hx) = sin(hx) = exp(x) - exp(-x) = exp(-i*ix) - exp(i*ix)
=> expression = -(sin(hx)) = sin(ix)/i
which is true!

arunmit168, what is cos(hx)? hyperbolic function? what is expression? what's the difference, the integral and the expression? one is the other's negative. i don't know your meaning.

Click to expand...

yes. cos(hx) is hyperbolic. its expression is same as that of Cos without the "j" inside it.. (for understanding).. that s,

cos(hx) = exp(x) + exp(-x) /2

hope i have written it there.. and there s no difference i see between the one u have given and that of mine.. the expression u used it direectly... is the one i have derived.. that is.. cos(ix) = cos(hx).. which i have proved and u have used!

similarly.. for sin(hx) = sin(ix) * i good luck!

/cedance