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# Prove some properties of the integral logarithm, Li (x)

The integral logarithm is defined for by Prove the following properties of .

1. .
2. where is a constant depending on . Find the value of for each .
3. Prove there exists a constant such that and find the value of this constant.

4. Let . Find an expression for in terms of .

5. Define a function for by Prove that 1. Proof. We derive this by integrating by parts. Let Then we have 2. Proof. The proof is by induction. Starting with part (a) we have To evaluate the integral in this expression we integrate by parts with This gives us Therefore we have where . This is the case . Now, assume the formula hold for some integer . Then we have We then evaluate the integral in this expression using integration by parts, as before, let Therefore, we have Plugging this back into the expression we had from the induction hypothesis we obtain Therefore, the formula holds for the case , and hence, for all integers , where  and make the substitution , . This gives us . Therefore, where is a constant 4. (Note: In the comments, tom correctly suggests an easier way to do this is to use part (c) along with translation and expansion/contraction of the integral. The way I have here works also, but requires an inspired choice of substitution.) We start with the given integral, and make the substitution Therefore, using the given fact that , we have 5. From part (d) we know that Then, for the term we consider the integral where . Similar to part (d) we make the substitution, This gives us Therefore, we have Taking the derivative we then have 