Page 879 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 859

where
1 2 n 2 m
a nm = (1 − (−1) ) (1 − (−1) ),
sinh(π n π + m /4) nπ mπ
2
2
2
8 1 − (−1) n 1 − (−1) m
c nm = √
2
2
2
sinh(2π n π + m ) nπ mπ
Section 18.7 Steady-State Equation for a Sphere
ρ n
2
1. u(ρ,ϕ) = ∞ (2n + 1)A 1 (arccos(ξ)) P n (ξ) P n (cos(ϕ))
n=0
2 −1 R
ρ ρ
2
≈ 2.9348A − 3.7011A P 1 (cos(ϕ)) + 1.1111A P 2 (cos(ϕ))
R R
ρ ρ
3 4
−0.5397A P 3 (cos(ϕ)) + 0.3200A P 4 (cos(ϕ))
R R
ρ 5
−0.2120A P 5 (cos(ϕ)) +···
R
ρ
3. u(ρ,ϕ) ≈ 6.0784 − 9.8602 P 1 (cos(ϕ))
R
ρ ρ
2 3
+5.2360 P 2 (cos(ϕ)) − 2.4044 P 3 (cos(ϕ))
R R
ρ ρ
4 5
+1.5080 P 4 (cos(ϕ)) − 0.9783 P 5 (cos(ϕ)) + ···
R R

T 1 R 1 1
5. u(ρ,ϕ) = R 2 − 1
R 2 − R 1 ρ
Section 18.8 The Neumann Problem
4
1. u(x, y) = c 0 − cosh(π(1 − y))cos(πx)
π sinh(π)
n
cosh(3(π − y)) ∞ 12((−1) − 1)
3. u(x, y) = c 0 − cos(3x) + n=1 cosh(ny)cos(nx)
3
3sinh(π) n π sinh(nπ)
2
2
n
2
n
∞
5. u(x, y) = [n π (−1) + 6(1 − (−1) )]cosh(nπ(1 − x))sin(nπy)
n=1 4 4
n π sinh(nπ)
1 R r 2
7. u(r,θ) = a 0 + cos(2θ)
2 2 R
2
2
1 ∞
9. u(x, y) = ln(y + 9(ξ − y) )ξe −|ξ| sin(ξ)dξ
2π −∞
11.
∞

u(x, y) = a ω cos(ωx)e −ωy dω
0
with
2 ∞
a ω =− f (ξ)cos(ωξ)dξ
πω 0
CHAPTER NINETEEN COMPLEX NUMBERS AND FUNCTIONS
Section 19.1 Geometry and Arithmetic of Complex Numbers
1. 26 − 18i 3. (1 + 18i)/65 5. 4 + 228i 7. 6 − i
9. (−1632 + 2024i)/4225 11. π/2 + 2kπ, k any integer; |3i|= 3
√
13. |− 3 + 2i|= 13; arctan(−2/3) + π + 2kπ 15. |− 4|= 4; π + 2kπ
√ √
17. 2 2e 3πi/4 19. 29e i arctan(−2/5)
√
21. 65e i(arctan(1/8))
2n
2 2n
4n
23. i = (i ) = (−1) = 1,i 4n+1 = i, i 4n+2 =−1, i 4n+3 =−i
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October 14, 2010 17:50 THM/NEIL Page-859 27410_25_Ans_p801-866