Ÿ“Œ‹ž“s@‚Ï‚¸‚«‚¿ ‚³‚ñ‚©‚ç‚̉ð“šB
y–â‘è‚Pz
2n.cos(ƒ¿1)cos(ƒ¿2)...cos(ƒ¿n)
| n ƒ® k=1 | { Exp(i ƒ¿k) + Exp(-i ƒ¿k) } |
| 1 ƒ° s1=0 |
1 ƒ° s2=0 | ... | 1 ƒ° sn=0 | Exp{i | n ƒ° k=1 |
(-1)sk ƒ¿k) } |
| 1 ƒ° s1=0 |
1 ƒ° s2=0 | ... | 1 ƒ° sn=0 | Exp{-i | n ƒ° k=1 |
(-1)sk ƒ¿k) } |
| 1 ƒ° s1=0 |
1 ƒ° s2=0 | ... | 1 ƒ° sn=0 | Cos{ | n ƒ° k=1 |
(-1)sk ƒ¿k) } |
‚±‚±‚Å
| r = | n ƒ° k=1 |
sk@‚Æ’u‚« |
| n ƒ° k=1 |
((-1)sk ƒ¿k)@= A-2(ƒ¿i1+ƒ¿i2+...+ƒ¿ir) |
| 1 ƒ° s1=0 |
1 ƒ° s2=0 | ... | 1 ƒ° sn=0 | ¨ | n ƒ° r=0 | ƒ° i1<i2<...<ir |
‚Æ‚È‚éB‚æ‚Á‚ÄAŽ®‚ªØ–¾‚³‚ꂽB
y–â‘è‚Qz
2-1)
xn - yn
| = | n-1 ƒ® k=0 | {x - Exp( | -i2ƒÎk n |
) y} |
| =i-n+1 | n-1 ƒ® k=0 | {Exp( | iƒÎk n |
) x - Exp( | -iƒÎk n |
) y } |
2-2)
xn + yn
| = | n-1 ƒ® k=0 | {x - Exp( | -iƒÎ(2k+1) n |
) y} |
| =i-n | n-1 ƒ® k=0 | {Exp( | iƒÎ(2k+1) 2n |
) x - Exp( | -iƒÎ(2k+1) 2n |
) y } |
y–â‘è‚Rz
3-1j
—^Ž®‚Í@(y+i)n - (y-i)n = 0‚Æ•ÏŒ`‚Å‚«‚é‚Ì‚Å
| y{i = Exp( | i2ƒÎk n |
) (y-i)@@ (k = 1,....n-1) |
| y = i(Exp( | i2ƒÎk n |
)+1)/(Exp( | i2ƒÎk n |
)-1)= cot( | ƒÎk n |
) |
3-2j
—^Ž®‚Í(y+i)n + (y-i)n = 0‚Æ•ÏŒ`‚Å‚«‚é‚Ì‚Å
| y{i = Exp(iƒÎ | 2k+1 n |
) (y-i)@@ (k =0, 1,....n-1) |
| y =cot( | ƒÎ(2k+1) 2n |
) |
y‚¨‚Ü‚¯z
| ‡ ƒ° n=1 | 1 22n+1¥n | 2nCn |
| = | ‡ ƒ° n=1 | 1 2n |
(2n-1)!! (2n)!! |
| = | ‡ ƒ° n=1 |
1 ç 0 |
(2n-1)!! (2n)!! | x2n-1dx |
| = | 1 ç 0 |
{(1-x2)(-1/2) - 1}/x} dx |
| = [-ln{1 +(1-x2)(1/2)} ] | 1 @ 0 |
| = ln2 |
‚½‚¾‚µ
| (2n-1)!! = (2n-1)*(2n-3)*.....*3*1 = | (2n)! (2n)!! |
Ž„‚Í{(1-x2)(-1/2) - 1}/x‚Ì•s’èÏ•ª‚ª’¼‚®‚É‚í‚©‚ç‚È‚©‚Á‚½‚Ì‚Å
x=sin(t)‚Æ’u‚¢‚Ä
| 1 ç 0 |
{(1-x2)(-1/2) - 1}/x}dx |
| = | ƒÎ/2 ç 0 |
{(cos(t))(-1) - 1}/sin(t)} cos(t) dt |
| = | ƒÎ/2 ç 0 |
tan(t/2) dt |
| = [-2 ln{ cos(t/2) }] | ƒÎ/2 @ 0 |
‚Æ‚µ‚Ä‹‚ß‚Ü‚µ‚½B