Relationship Between Trig Funtions and Bernoulli Numbers

In summary, to prove the formula xcsc(x)=2B(ix)-B(2ix), one can use the formulas B(x)=x/((e^x)-1) and sin(x)=(e^{ix}-e^{-ix})/2i. By manipulating these equations and using the difference of squares, it can be shown that x/sin(x) is equal to (2ix)/(e^{ix}-e^{-ix})-x/((e^{2ix}-1)).
  • #1
lwest513
1
0

Homework Statement


Prove the formula xcscx=2B(ix)-B(2ix)

Homework Equations


B(x)=x/((ex)-1)

sinx= (eix-e-ix)/2i

The Attempt at a Solution



I know that it makes sense to use the formula for B(x) with x=ix and x=2ix, and rewrite xcsc(x) as x/sin(x), plugging the above relevant equation in for sine. Manipulating these equations should bring about equality, but for some reason I've hit a road block where I can't come up with more algebraic manipulation to get to the equality.

I have that xcsc(x)= (2ix)/(eix-e-ix)
but after expanding 2B(ix)-B(2ix) using the B(x) formulas, I'm having a difficult time figuring out how to reduce it correctly and I'm too stuck in the same methods to see it a different way.
Right now I have (2ix(e2ix-eix))/(e3ix-eix-e2ix+1)

Any help would be greatly appreciated because I'm too caught up in a method that clearly isn't working!

EDIT:I figured it out but I'm not sure how to delete the post so I'll leave it. If anyone else is having the same problem, the key is to use the difference of squares on the B(2ix) term denominator.
 
Last edited:
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  • #2
hint factor two different ways

$$(e^{i x}-1)(e^{i x}+1)=(e^{2i x}-1)=e^{ix}(e^{i x}-e^{-i x})$$
 

Related to Relationship Between Trig Funtions and Bernoulli Numbers

1. What is the relationship between trigonometric functions and Bernoulli numbers?

The relationship between trigonometric functions and Bernoulli numbers can be seen in the Euler-Maclaurin summation formula, which uses Bernoulli numbers to approximate the sum of a series. Additionally, the periodicity of trigonometric functions is related to the periodicity of Bernoulli numbers.

2. How are Bernoulli numbers used in trigonometry?

Bernoulli numbers are used in trigonometry to approximate the sums of infinite series, such as those found in Fourier series. They are also used in the Euler-Maclaurin summation formula to calculate the error in approximating a series using finite sums.

3. Can trigonometric functions be expressed in terms of Bernoulli numbers?

Yes, trigonometric functions can be expressed in terms of Bernoulli numbers using the Euler-Maclaurin summation formula. This allows for the calculation of the values of trigonometric functions at non-standard angles.

4. Are there any other applications of the relationship between trigonometric functions and Bernoulli numbers?

Yes, there are many other applications of this relationship. For example, Bernoulli numbers are used in the study of elliptic curves, which have applications in cryptography. They are also used in the calculation of various infinite series and in the proof of mathematical theorems.

5. How did the relationship between trigonometric functions and Bernoulli numbers come about?

The relationship between trigonometric functions and Bernoulli numbers was first studied by the mathematician Leonhard Euler in the 18th century. He noticed the connection between the periodicity of trigonometric functions and the periodicity of Bernoulli numbers, and developed the Euler-Maclaurin summation formula to further explore this relationship.

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