How to find the argument of a function?

AI Thread Summary
To determine the argument of the function V(f) = A τ sinc(f τ), it is essential to analyze the behavior of the sinc function, particularly its transitions between positive and negative values. The argument can be concluded as 0º, +180º, or -180º based on the phase shifts that occur when the sinc function crosses zero. The relationship between the imaginary and real components, as defined by arg = arctan(Im/Re), helps clarify these phase shifts. Understanding the amplitude A, which indicates the function's value at x = 0, is also crucial for determining the overall behavior of the function. Visualizing the function through diagrams can further aid in grasping these concepts.
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Homework Statement


IMG_20150601_210114.jpg

For the function ##V(f) = A \tau sinc(f \tau)## in the picture how do you conclude that the argument of ##V(f)## is 0º, +180º or -180º?

Homework Equations


##arg = arctan({\frac{Im}{Re}})##

The Attempt at a Solution


The upper graphic represents the absolute value of the function, right? I understand that being a sinc function the negative parts of the graphic should "jump" to the positive side. I also understand why the function return 0 on k/τ.
Is it correct to think, for example, between 1/τ and 2/τ the sinc function returns negative values but because it needs to shift to the positive values the fase should be negative so that negative times negative = positive?
I'm so sorry if this questions sounds too basic. I'm trying to learn by myself, I'm not attending any classes and I have no one to ask this kind of doubts.
Thanks in advance.
 
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Write analytical ##sinc()## function and think about ##sin()## that you know much better.
 
##sinc(x) = {\frac{\sin(\pi x)}{\pi x}}##

Capture.jpg

The ##sin()## is positive between ##-2 \pi## and ##- \pi## and between ##0## and ##\pi## and is negative between ##- \pi## and ##0## and between ##\pi## and ##2 \pi##...
 
On your function the argument is ##ft## so ##f=\pi##. What must be ##Α## in your function?
 
##A## represents the amplitude of the function, the value that the function return when ##x = 0##
 
I think ##A## has only a sign. The second diagram is a step function. Make some diagrams.
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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