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Derive 13.4 to 13.6 (Arfken's approach) please show complete solution

Derive 13.4 to 13.6 (Arfken's approach) please show complete solution

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Derive 13.4 to 13.6 (Arfken's approach) please show complete solution
Gamma Function (Chapter 13) To set up the properties of the gamma function, the following definitions must be used: 1) Infinite limit (Euler) I(z) = Consider z→→z+1 1.2.3. n nco (z + 1)(z+2)(z + 3) (z+1+n) 1.2.3... n nz = lim -n² n→∞ (z+1+n) z(z + 1)(z + 2)(z + 3). (z+n)' T(z + 1) = lim Arfken, G., Weber, H., and Harris, F. (2013). Mathematical Methods for Physicists (7 od.). Elsevier. 1.2.3...n lim n→∞ z(z + 1)(z+2)... (z+n) z 0,-1, -2, -3,... (13.1) r(1) = lim In general, I(z + 1) nz 1.2.3...n = lim -n² no (z+1+n) z(z + 1)(z + 2)(z + 3) (z+n) = zl(z) (13.2) From the definition I(z) = 1.2.3...n Ariken, G., Weber, H., and Harris, F. (2013). Mathematical Methods for Physicists (7a ed.). Elsevier. 2) Definite integral (Euler) no 1.2.3 n(n+1) r(2) = 1 r(3) = 2r(2) = 2 r(4) = 3r(2) = 3·2=6 so on. I(n) = 1.2.3 (n-1)=(n-1)! (13.4). r(z) = 2 = 00 Transform using t = u² and dt = 2u du, = - e-u²u²z-2 2udu n = 1 -n², [(z) = e-t²-1 dt, Re(z) 0 (13.5). 00 ₂-u², ²u²z-1 du, e (13.3) Re(z) > 0 -n²+1 (13.6) 10
Transcribed Image Text:Gamma Function (Chapter 13) To set up the properties of the gamma function, the following definitions must be used: 1) Infinite limit (Euler) I(z) = Consider z→→z+1 1.2.3. n nco (z + 1)(z+2)(z + 3) (z+1+n) 1.2.3... n nz = lim -n² n→∞ (z+1+n) z(z + 1)(z + 2)(z + 3). (z+n)' T(z + 1) = lim Arfken, G., Weber, H., and Harris, F. (2013). Mathematical Methods for Physicists (7 od.). Elsevier. 1.2.3...n lim n→∞ z(z + 1)(z+2)... (z+n) z 0,-1, -2, -3,... (13.1) r(1) = lim In general, I(z + 1) nz 1.2.3...n = lim -n² no (z+1+n) z(z + 1)(z + 2)(z + 3) (z+n) = zl(z) (13.2) From the definition I(z) = 1.2.3...n Ariken, G., Weber, H., and Harris, F. (2013). Mathematical Methods for Physicists (7a ed.). Elsevier. 2) Definite integral (Euler) no 1.2.3 n(n+1) r(2) = 1 r(3) = 2r(2) = 2 r(4) = 3r(2) = 3·2=6 so on. I(n) = 1.2.3 (n-1)=(n-1)! (13.4). r(z) = 2 = 00 Transform using t = u² and dt = 2u du, = - e-u²u²z-2 2udu n = 1 -n², [(z) = e-t²-1 dt, Re(z) 0 (13.5). 00 ₂-u², ²u²z-1 du, e (13.3) Re(z) > 0 -n²+1 (13.6) 10
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