## Nielsen and Chuang Exercise 2.35

Exercise 2.35Exponential of the Pauli matrices: Let be any real three-dimensional unit vector, and a real number. Prove that

We will break the integral into , , and components, and note that the eigenvalues of the Pauli matrices are (-1,1). Each component is then given by:

with analogous expressions for and . We can use the standard expansion of the exponent

We note that the cosine is an even function, and the sine is an odd function, so that this term reduces to

First, note that because

we can show that

and

Plugging this into the Taylor expansion for (which we will break into even and odd powers) gives

If anyone has any suggestions on how to finish my original proof, let me know.

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