Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 3 pdf

... x2, x 3 , x 4 } is independent, but not orthogonal in the interval [−1, 1]. Using Gramm-Schmidt orthogo-12 84 12 3 4 56-60 -40 -20Figure 24. 3: log(error in approximation)In Figure 24. 4 we ... =1√πx−1e−x2−1√π∞xt−2e−t2dt=1√πx−1e−x2−1√π−12t 3 e−t2∞x+1√π∞x 3 2t 4 e−t2dt=1√πe−x2x−1−12x 3 +1√π∞x 3 2t 4 e−t2dt=1√πe−x2x−1−12x 3 +1√π− 3 4 t−5e−t2∞x−1√π∞x15 4 t−6e−t2dt=1√πe−x2x−1−12x 3 + 3 4 x−5−1√π∞x15 4 t−6e−t2dt1265The ... 21)π 4 P 4 (x)=1058π 4 [ (31 5 − 30 π2)x 4 + ( 24 2− 270)x2+ (27 − 2π2)]The cosine and this polynomial are plotted in the second graph in Figure 25.1. The le ast squares fit method usesinformation...

... solution and discuss in as much detail as possible what goes wrong. 13 64 Note that this formula is valid for m = 0, 1, 2, . . Similarly, we can multiply by sin(mx) and integrate to solve for bm. ... +n2x2− 2n 3 sin(nx) 137 0 -3 -2-1 12 3 -3 -2-112 3 Figure 28.2: Graph ofˆf(x).would give a better approximation?Least squared error fit. The most common criterion for finding the best ... 21/20x sin(nπx) dx + 211/2(1 − x) sin(nπx) dx= 4 (nπ)2sin(nπ/2)= 4 (nπ)2(−1)(n−1)/2 for odd n0 for even n. 135 6 -3 -2-1 12 3 -1.5-1-0.50.511.5Figure 28.5: Fourier Sine Series.The...

... . . . . . . . . . . . . . . 1 630 34 . 3. 3 Bessel Functions of Non-Integer Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 633 34 . 3 .4 Recursion Formulas . . . . . . . . . . . ... . . 1910 43 .10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1911 44 Transform Methods 1918 44 .1 Fourier Transform for Partial Differential ... . . . . . 147 2 30 .3. 2 Singular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3 31 The Laplace Transform 147 5 31 .1 The Laplace Transform . . ....