[理工] [工數]複變實變函數定積分
請教各位高手複變幾個小問題
1.
2π dθ
Evaluate ∫ ---------
0 5+3sinθ
sol:
iθ dz 1 -1
令C為|z| = 1，則z = e ，dθ = ---- 且sinθ = ----(z－z )
iz 2i
2dz
I = ............. = ∮ ----------------
c 3z^2 + 10iz - 3
2
令 f(z) = ---------------
3z^2 + 10iz - 3
故f(z)在C內具有 z = -i/3的單極點，且殘值為
(書上寫)
1 2 1
Resf(-i/3) = lim [(z + ---i) -----------------] = ----
z->-i/3 3 3z^2 + 10iz - 3 4i
(我的計算)
1 2
Resf(-i/3) = lim [(z + ---i) ------------------]
z->-i/3 3 1
(z + ---i)(z+3i)
1 3
然後消掉z + ---i代入z=-i/3
3
2 3
Resf(-i/3) = --------------------- = ----
-i 4i
--- + 3i
3
想請問我的計算是哪裡出錯了，想了兩天一直鬼打牆，煩請板上高手解惑
感謝解答!!
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