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In summary, the conversation discusses a problem in E&M where solving an equation depends on rewriting sinh(\sqrt{p^2 + k^2} \pi) as a product of two functions. The speaker suggests trying to use the Taylor expansion of sinh, but doubts it can be split in that way due to it being a transcendental equation. They also mention the possibility of separating it at an earlier stage of the problem.

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I think if you look at an earlier stge of the problem, you may be able to separate there.

To write sinh(...) as a product of p and k, you can use the identity sinh(x) = (e^x - e^-x)/2. Then, you can factor out a k from the numerator to get k(e^x - e^-x)/2. Finally, you can use the identity e^x - e^-x = 2sinh(x) to get the final form of p*sinh(x), where p = k/2.

Yes, in addition to the identity mentioned above, there are other identities that can be used to write sinh(...) as a product of p and k. These include sinh(x) = (e^x + e^-x)/2 and sinh(x) = (e^2x - 1)/(2e^x).

Writing sinh(...) as a product of p and k can be useful when solving E&M problems because it allows you to simplify equations and make them easier to solve. It also helps to identify the role of p and k in the equation and their individual contributions to the overall solution.

No, sinh(...) can only be written as a product of p and k for specific values of x. This is because sinh(...) is a function that is defined only for real numbers, so the identity used to write it as a product may not hold for all values of x.

Yes, there is a general formula for writing sinh(...) as a product of p and k. It is sinh(x) = p*sinh(kx), where p and k are constants. However, the specific values of p and k will vary depending on the identity used and the given equation.

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