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UnkleRhaukus

  • 2 years ago

question about notation

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  1. UnkleRhaukus
    • 2 years ago
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    in my book there is a question involving a term like this \[\int\limits_p^\infty F(p)\cdot\text dp \], but this is a confusing choice of notation because the variable of integration is one one of limits of integration

  2. UnkleRhaukus
    • 2 years ago
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    as far as i know the convention is to change to \[\int\limits_p^\infty F(p')\cdot\text dp'\] but i dont like this because it looks like the prime of p

  3. ash2326
    • 2 years ago
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    As we know that integration is independent of variable. You could use any variable of your choice. I'd go with the first one :)

  4. UnkleRhaukus
    • 2 years ago
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    a?

  5. ash2326
    • 2 years ago
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    yes

  6. UnkleRhaukus
    • 2 years ago
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    \[\int\limits_p^\infty F(a)\cdot\text da\]

  7. mukushla
    • 2 years ago
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    \[\int\limits_p^\infty F(x)\cdot\text dx\]

  8. waterineyes
    • 2 years ago
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    \[\int\limits\limits_p^\infty F(mukushla)\cdot\text d(mukushla)\]

  9. UnkleRhaukus
    • 2 years ago
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  10. UnkleRhaukus
    • 2 years ago
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    the notation is still troubling me

  11. waterineyes
    • 2 years ago
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    Where???

  12. UnkleRhaukus
    • 2 years ago
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    well i kinda had to change the question so i could answer it

  13. UnkleRhaukus
    • 2 years ago
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    \[\begin{align*} \int\limits_p^\infty F(q)\cdot\text dq &=\int\limits_p^\infty\int\limits_0^\infty f(x)e^{-qx}\cdot\text dx\cdot\text dq \\&=\int\limits_0^\infty f(x)\int\limits_p^\infty e^{-qx}\cdot\text dq\cdot\text dx \\&=\int\limits_0^\infty f(x)\left.\frac{e^{-qx}}{-x}\right|_p^\infty \cdot\text dx \\&=\int\limits_0^\infty \frac{f(x)}xe^{-px}\cdot\text dx \\&=\mathcal L\left\{\frac {f(x) }x \right\} \end{align*}\]

  14. UnkleRhaukus
    • 2 years ago
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    do you understand why i am getting confused/

  15. waterineyes
    • 2 years ago
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    The things you are doing are going above my head, but I am not understanding why are you getting confused, where are you not feeling comfortable??

  16. UnkleRhaukus
    • 2 years ago
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    well the question in the book makes no sense

  17. UnkleRhaukus
    • 2 years ago
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    @sasogeek

  18. sasogeek
    • 2 years ago
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    certainly this is confusing, but why not use u-substitution for F(p) ?

  19. UnkleRhaukus
    • 2 years ago
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    like this? \[\begin{align*} \int\limits_p^\infty F(p)\cdot\text dp\\ \text{let }u=p\\ \text du=\text dp\\ \\p=p\rightarrow u=u\\ p=\infty\rightarrow u=\infty\\ \\&=\int\limits_u^\infty F(u)\cdot\text du\\ \end{align*}\]

  20. sasogeek
    • 2 years ago
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    \(let \ u=F(p)\) \(\large \frac{du}{dp}=F'(p)\) \(\large dp =\frac{du}{F'(p)} \) \[\int\limits_{p}^{\infty}u \ \frac{du}{F'(p)}\]

  21. UnkleRhaukus
    • 2 years ago
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    im not sure why you'd do that?

  22. sasogeek
    • 2 years ago
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    well having the function variable being the same as one of the limits... i'm not quite sure how you'd go about solving it but thats the first thing that comes to my mind. besides, i'm not so good at calculus, but that's what i'd do. it may or may not be right, but it's worth a shot :/. I'm pretty certain this isn't calculus 1... right?

  23. UnkleRhaukus
    • 2 years ago
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    differential equations; laplace transforms

  24. sasogeek
    • 2 years ago
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    there we go, that's beyond me lol, but all things being equal, thats what i'd do. sorry i wasn't of much help lol :/ but i'll stick around to learn :)

  25. UnkleRhaukus
    • 2 years ago
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    but the problem im having is not the content of the question, i think i have answered that satisfactorily , my issus is the choice of notation used in the question

  26. UnkleRhaukus
    • 2 years ago
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    i dont think the question makes sense as written \[....=\int\limits_p^\infty F(p)\text dp\] it dosent make sense to have the variable as a limit of integration

  27. sasogeek
    • 2 years ago
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    you could ask satellite73 or amistre when they get online....

  28. UnkleRhaukus
    • 2 years ago
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    @amistre64

  29. amistre64
    • 2 years ago
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    yeah, i looked at this earlier and dont have anything pertinent to add to it :/ srry

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