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Thread: The Nth Derivative of Trigonometric Functions

  1. #1 The Nth Derivative of Trigonometric Functions 
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    In addition to being transcendental, trigonometric functions are classified as "periodic." Mathematicians classify them as such a class of function because any value that appears once in a trigonometric function will appear an infinite number of times. I.e., there are an infinite number of ways to demonstrate that the trigonometric functions are not bijective.

    The derivatives of trigonometric functions are also trigonometric functions, therefore their derivatives must also be periodic. Let us find the periodicity of the sine function to begin (defining the sine function as y = sin(x)):











    Therefore, from the above we can find the nth derivative of the sine function by taking a multiplier of four away consistently until our derivative is between one and four: that is to say, in algebraic terms, n - 4x.

    ,

    Let us do the same for the cosine function:










    ,


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  3. #2 Re: The Nth Derivative of Trigonometric Functions 
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    Quote Originally Posted by Ellatha
    In addition to being transcendental, trigonometric functions are classified as "periodic." Mathematicians classify them as such a class of function because any value that appears once in a trigonometric function will appear an infinite number of times. I.e., there are an infinite number of ways to demonstrate that the trigonometric functions are not bijective.
    is not periodic, but would be if your definition were adopted.



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  4. #3 Re: The Nth Derivative of Trigonometric Functions 
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    Quote Originally Posted by DrRocket
    is not periodic, but would be if your definition were adopted
    Well, I should have said that not all trigonometric functions are periodic. Specifically, I was referring to the sine and cosine functions (as these were the two that I demonstrated).

    Correction: a trigonometric function is periodic if its output values appear over a set interval; although all trigonometric functions repeat themselves infinitely, not all occur over regular intervals.
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  5. #4 Re: The Nth Derivative of Trigonometric Functions 
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    Quote Originally Posted by Ellatha
    Quote Originally Posted by DrRocket
    is not periodic, but would be if your definition were adopted
    Well, I should have said that not all trigonometric functions are periodic. Specifically, I was referring to the sine and cosine functions (as these were the two that I demonstrated).
    You missed the point. The problem is with your definition of periodic.

    A function is periodic, with period if .

    This is easily extended to functions defined on an abelian group, with arbitrary range, and to left or right periodic functions on an arbitrary group.

    Also is not what one normally considers to be a trigonometric function. It is an example of a function that meets your criteria but is not periodic.
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  6. #5  
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    I was aware that you were saying that my definition of periodicity is incorrect: I have fixed the definition above (I should have noted that the function must repeat over a fixed interval [which is more intuitive than the definition that you provided]).
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  7. #6  
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    An example of the application of the formulas derived by the above proofs:



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  8. #7  
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    Quote Originally Posted by Ellatha
    An example of the application of the formulas derived by above proofs:



    What is more important is that is an eigenfunction for . That fact explains much of the usefullness of Fourier analysis.
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  9. #8  
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    Quote Originally Posted by DrRocket
    What is more important is that is an eigenfunction for . That fact explains much of the usefullness of Fourier analysis.
    I didn't know the thread had some relation to the Fourier series; I thought I was merely showing the periodic nature of the derivative of the sine and cosine functions.

    By the way, if we define a periodic function as f(x) = f(x + c), than can we say the following:







    if , than

    In other words, the derivative of any periodic function is periodic (meaning the nth derivative of a periodic function is also periodic, as can be demonstrated via mathematical induction).

    The above proof assumes that x is an element of the real numbers.*
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  10. #9  
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    Quote Originally Posted by Ellatha
    I didn't know the thread had some relation to the Fourier series;
    Not just Fourier series. The Fourier transform also.
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  11. #10  
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    Quote Originally Posted by DrRocket
    Not just Fourier series. The Fourier transform also.
    What about my proof?
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  12. #11  
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    Quote Originally Posted by Ellatha
    What about my proof?
    You don't want to know.
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  13. #12  
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    Quote Originally Posted by DrRocket
    You don't want to know.
    Yes I do.





    if , than
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  14. #13  
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    Quote Originally Posted by Ellatha
    Quote Originally Posted by DrRocket
    You don't want to know.
    Yes I do.





    if , than
    OK, but remember that you insisted.

    The proofs of both the derivative if the translate of a function an the higher-order derivatives of the sine and cosine are correct. But they are so trivial that there is little point in even writing them down. They would be just a remark in a mathematics text.
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  15. #14  
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    Quote Originally Posted by DrRocket
    OK, but remember that you insisted.

    The proofs of both the derivative if the translate of a function an the higher-order derivatives of the sine and cosine are correct. But they are so trivial that there is little point in even writing them down. They would be just a remark in a mathematics text.
    That is fine--I wasn't trying to make an important contribution to mathematics; I was trying to make a contribution to the forum members. I haven't the knowledge yet to understand the important, open mathematical questions so that I may potentially find a solution. Maybe that will change someday.
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