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Let denote one of the Jacobi Theta functions. Prove that
We have successively,
The sum is evaluated as follows. Consider the function
and integrate it around a square with vertices . The function has poles at every integer with residue as well as at with residues . We also note that as the contour integral of tends to . Thus,
and the exercise is complete.
Prove that forall and it holds that
This is a very classic exercise and can be dealt with various ways. We know the result in advance. Why? Because it is the Taylor Polynomial of the exponential function. Let us see however how we gonna deal with it with High School Methods.
Find the positive real number such that
Define the function
and note that forall . Clearly , is differentiable and its derivative is given by
It follows that . Suppose that . Then the monotony of as well as the sign of is seen at the following table.
It follows then that . This is an obscurity due to the fact that . Similarly, if we suppose that . Hence
For we easily see that the given inequality holds.
Let be a twice differentiable function such that forall . Prove that:
The equation of the tangent of at is
Since forall we deduce that is strictly convex. Thus, the tangent lies below the graph of except at . Hence,
and the conclusion follows.