acase@tulane.edu

acase@tulane.edu

A Landen transformation is a transformation on the parameters of a definite integral that fixes its value. The earliest discovery of such a transformation was the arithmetic-geometric mean iteration, phi: (a, b) ((a + b)/2, ab ). Gauss connected the limit of this iteration to the complete elliptic integral, whose value is fixed under phi. This result both links the study of integrals and Number Theory and provides a quadratically convergent numerical method for approximating an elliptic integral We present a Landen transformation for a rational integral over the real line. This generalizes the work done by Boros and Moll for the even case. Let Rp :=&cubl0;a&ar;e R2p&vbm0;I a&ar;< infinity&cubr0;, where a&ar;:= a0,&ldots;,ap;b0,&ldots;,b p-2and Ia&ar; := RRx dx , with Rx :=BxA x=b0xp -2+&cdots;+bp-2a0xp +a1xp-1+&cdots;+ap . We construct, for all 2 &le; m, p &isin; N , a rational Landen transformation Lm,p :Rp Rp , and prove its scaled version converges to a limit in R2p with convergence order m. We also prove the invariant property Ia&ar; =I&parl0;Lm,p a&ar; &parr0;, which leads to the formula RR xdx=pLa &d1;, where La&ar; :=lim n&rarr;infinitybn 0an 0, Lnm, na =:&parl0;a n0,a n1,&cdots;,a np;b n0,&cdots;b np-2&parr0; is the limit of the Landen iteration. This generates a numerical method for a rational integral which converges to order m / acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

The thesis argues that we arrive at a more coherent explanation of Kant's thought on the basis of the premise that he is primarily concerned with moral issues. This approach contrasts many of Kant's commentators in the English speaking world who believe that he is mostly concerned with demonstrating the limits of reason. I show that Kant's critical and practical works together explain the conditions of moral experience, and argue that throughout his works Kant attempts to motivate the subject to act on the basis of a pure moral incentive as well as continually strive for moral betterment. Additionally, I show how for Kant, the single individual inwardly experiences moral goodness There are two aspects of Kant's thought which have not been paid adequate attention to which the thesis addresses. The first concerns the relationship between theoretical reason's demand for an unconditioned condition and the practical philosophy, Kant tells us that the theoretical philosophy is unable to satisfy the demands of reason for an account of an unconditioned condition, but that the practical philosophy is able to do so. The thesis therefore analyzes the relationship between the unconditioned condition and moral experience. Practical reason discovers that the free will is an unconditioned condition as the ground of action of man in the word, and at the same time it constructs an unconditioned condition in the form of the object the good will strives after, namely the Kingdom of God on earth The second essential and primary argument of the thesis is that Kant is articulating the specifically human mode of the experience of morality. This mode is not one of attaining a good will once and for all, but rather one of continual becoming. Kant tells us that we must continually strive for moral betterment while doing so on the basis of a pure incentive. In summary, while Kant is usually considered one of the forerunners of modern philosophy's skepticism, I argue that he is better understood as telling us that the highest end of philosophy is not knowledge of objects but the moral betterment of humanity / acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu