| x=aの周りの展開 |
f(x)=f(a)+f'(a)(x-a)/1!+f"(a)(x-a)2/2!+f(3)(a) (x-a)3/3!+...+f(n)(a)(x-a)n/n!+O((x-a)n+1) |
| ex, x=1 | (1/e){1+(x-1)+(x-1)2/2+(x-1)3/3!+(x-1)4/4!+(x-1)5/5!+(x-1)6/6! +(x-1)7/7!+(x-1)8/8!+(x-1)9/9!+(x-1)10/10!}+O((x-1)11) |
| ln(x),x=1 |
f(x)=x-1-(x-1)2/2+(x-1)3/3-(x-1)4/4+(x-1)5/5-(x-1)6/6+(x-1)7/7-(x-1)8/8 +(x-1)9/9-(x-1)10/10+O(x11) |
| sin(x), x=π/4 |
f(x)={(√2)/2}{1+(x-(π/4))-(x-(π/4))2/2!-(x-(π/4))3/3!+(x-(π/4))4/4! +(x-(π/4))5/5!-(x-(π/4))6/6!-(x-(π/4))7/7!+O((x-(π/4))8) |
| sin(x), x=π/2 |
f(x)=1-(x-(π/2))2/2+(x-(π/2))4/24-(x-(π/2))6/720+(x-(π/2))8/40320-(x-(π/2))10/3628800+... |
| cos(x), x=π/4 |
f(x)=sqrt(2)/2-(sqrt(2)*(x-π/4))/2-sqrt(2)*(x-(π/4))2/4+sqrt(2)*(x-(π/4))3/12+sqrt(2)*(x-(π/4))4/48 -sqrt(2)*(x-(π/4))5/240-sqrt(2)*(x-(π/4))6/1440+sqrt(2)*(x-(π/4))7/10080+sqrt(2)*(x-(π/4))8/80640 -sqrt(2)*(x-(π/4))9/725760-sqrt(2)*(x-(π/4))10/7257600+... |
| cos((x), x=π/2 |
f(x)=-(x-π/2)+(x-π/2)3/6-(x-π/2)5/120+(x-π/2)7/5040-(x-π/2)9/362880+... |
| tan(x),x=π/4 |
f(x)=1+2*(x-π/4)+2*(x-π/4)2+8*(x-(π/4))3/3+10*(x-(π/4))4/3+ 64*(x-(π/4))5/15+244*(x-(π/4))6/45+2176*(x-(π/4))7/315+554*(x-(π/4))8/63+31744*(x-(π/4))9/2835+ 202084*(x-(π/4))10/14175+... |
| 求め方 |
[wxMaxima] (%i 1)
taylor(exp(x),x,1,10); (%i 2) taylor(log(x),x,1,10); (%i 3) taylor(log(1+x),x,0,10); (%i 4) taylor(tan(x),x,%_pi/4,10); [Maple] series(exp(x),x=a,10); |
[1]http://next1.msi.sk.shibaura-it.ac.jp/MULTIMEDIA/calc/node22.html
[2]
[3]
[4]
陰関数f(x,y)=0のテーラー展開
| f(x,y)=0 y=g(x) x=aの周りの展開 |
fx+fy*y'=0, y'=g'(x)=-fx/fy fxx+2fxy*y'+fxy(x,y)y'2+fy(x,y)y"=0 y"=g"(x)=-(fxx*fy2-2*fx*fxy*fy+fx2*fxy)/fy3 y'''=g'''(x)=-(fx(x,y)* fxx(x,y)*fy(x,y)2*fyy(x,y)-2*fx(x,y)2*fxy(x,y)*fy(x,y)*fyy(x,y)+fx(x,y)3*fxy(x,y)*fyy(x,y)+fxxx(x,y)* fy(x,y)4-3*fxx(x,y)*fxy(x,y)*fy(x,y)3-3*fx(x,y)*fxxy(x,y)*fy(x,y)3+2*fx(x,y)2*fxyy(x,y)*fy(x,y)2+6*fx(x,y)* fxy(x,y)2*fy(x,y)2+2*fx(x,y)*fxx(x,y)*fxy(x,y)*fy(x,y)2+fx(x,y)2*fxxy(x,y)*fy(x,y)2-fx(x,y)3*fxyy(x,y)* fy(x,y)-7*fx(x,y)2*fxy(x,y)2*fy(x,y)+2*fx(x,y)3*fxy(x,y)2)/fy(x,y)5 y=g(x)=g(a)+g'(a)(x-a)/1!+g"(a)(x-a)2/2!+g'''(a) (x-a)3/3!+...+g(n)(a)(x-a)n/n!+O((x-a)n+1) |
| [演習] f(x,y)=(x2+y2)^2-4xy=0 の (x,y)=(1,1) におけるテーラー展開を求めよ。 |
[解答] y=g(x)とおくと (%i1) diff(x^3-(x^2-y^2)/2-x*y^2=0); (%o1) (y-2*x*y)*del(y)+(-y^2+3*x^2-x)*del(x)=0 (y-2*x*y)*dydx(x,y)+(-y^2+3*x^2-x)=0 (x,y)=(1,1)とおくと -dydx(1,1)+1=0 , dydx(1,1)=1 |
二変数テーラー展開(Taylor series Expansion)：(x,y)=(a,b)の周りの展開
|
f(x,y)=f(a,b)+{fx(a,b)(x-a)+fy(a,b)}/1! +[fxx(a,b)(x-a)2+{fxy(a,b)+fyx(a,b)}(x-a)(x-b)+fyy(a,b)(y-b)2]/2! +[fxxx(a,b) (x-a)3+{fxxy(a,b)+fxyx(a,b)+fyxx(a,b)}x2y+{fxyy(a,b)+fyxy(a,b)+fyyx(a,b)}xy2 +fyyy(a,b)y3]/3!+...+{(x-a)Dx+(y-b)Dy}nf(a,b)/n!+... (x-a)2Dx2 f(a,b)=(x-a)2 fxx(a,b), (x-a)(y-b)DxDy(a,b)=(x-a)(y-b)fxy(a,b), etc |
| f(x,y) = 3x2+4xy-5y2の
(1,-2)のまわりでの ２次のテイラー展開を求めよ。 |
f(x,y)=f(1,-2)+(x-1)fx(1,-2)+(y+2)fy(1,-2) +(x-1)2(1/2)fxx(1,-2)+(y+2)2(1/2)fyy(1,-2)+(x-1)(y+2)fxy(1,-2) +R3 =-25-2(x-1)+24(y+2)+3(x-1)2-5(y+2)2+4(x-1)(y+2), R3=0 [Maple10] > f := 3 x2 - 5 y2 + 4 x y; eval(f, [x = 1, y = -2]); f1 := series(f, x = 1, 3); f2 := series(f, y = -2, 3); f:=3 x2 + 4 x y - 5 y2 -25 f1:= 3 - 5 y2 + 4 y + (4 y + 6) (x - 1) + 3 (x - 1)2 f2:= 3 x2 - 8 x - 20 + (20 + 4 x) (y + 2) - 5 (y + 2)2 > fx := ∂f/∂x; eval(fx, [x = 1, y = -2]); fxx :=∂fx/∂x; fx:= 6 x + 4 y -2 fxx: = 6 > fy := ∂f/∂y; eval(fy, [x = 1, y = -2]); fyy :=∂fy/∂y; fy:=4 x - 10 y 24 fyy:= -10 > fxy := ∂fx/∂y; fyx := ∂fy/∂x; eval(fxy, [x = 1, y = -2]); fxy:= 4 fyx= 4 4 > simplify(expand(-25 - 2 (x-1) + 24 (y+2) + 3 (x - 1)2 - 5 (y + 2)2 + 4 (x - 1) (y + 2))); s:= 3 x2 - 5 y2 + 4 x y > convert(s, 'string'); "3 x2-5 y2+4*x*y" |
| f(x,y) = exy
の(0,0)のまわりでの２次のテイラー展開を求め、剰余項R3の具体的な形を求めよ。 |
f(x,y)=f(0,0)+xfx(0,0)+yfy(0,0)+(1/2)x2fxx(0,0)+(1/2)y2fyy(0,0) +xyfxy(0,0)+R3 =1+xy+R3 f(x,y)=1+xy+R3, R3=exy-1-xy [Maple10]  exp(x y) 1 y exp(x y) 0 x exp(x y) 0 exp(x y) + y x exp(x y) 1 exp(x y) + y x exp(x y) 1 y2 exp(x y) 0 x2 exp(x y) 0 >   |
[1]http://markun.cs.shinshu-u.ac.jp/learn/biseki/no_9/cont09_3.html
[2]
[3]
マクローリン展開：テイラー展開でx=0の周りの展開
| x=aの周りの展開 |
f(x)=f(0)+f'(0) x/1!+f"(0) x2/2!+f(3)(0)
x3/3!+...+f(n)(0)(xn)/n!+O(xn+1)
(収束範囲) |
| ex, x=0 | 1+x+ x2/2!+x3/3!+x4/4!+x5/5!+x6/6!+x7/7!+x8/8!+x9/9!+x10/10!+O(x11)
(-1<x<1) |
| ln(1+x),x=0 | f(x)=x- x2/2+x3/3-x4/4+x5/5-x6/6+x7/7-x8/8+x9/9-x10/10+O(x11) (-1<x<1) |
| 1/(1+x), x=0 | f(x)=1-x+ x2-x3+x4-x5+x6-x7+x8-x9+x10+O(x11) (-1<x<1) |
| 1/(1-x), x=0 | f(x)=1+x+ x2+x3+x4+x5+x6+x7+x8+x9+x10+O(x11)
(-1<x<1) =Σ[n=0, ∞] xn (-1<x<1) |
| 1/(1+x2), x=0 | f(x)=1- x2+x4-x6+x8-x10+... =Σ[n=0, ∞] (-1)n x2n (-1<x<1) |
| 1/(1-x2), x=0 | f(x)=1+ x2+x4+x6+x8+x10+... =Σ[n=0, ∞] x2n (-1<x<1) |
| 1/√(1+x), x=0 | f(x)=1-x/2+3* x2/8-5*x3/16+35*x4/128-(63*x5)/256+(231*x6)/1024 -(429*x7)/2048+(6435*x8)/32768-(12155*x9)/65536+(46189*x10)/262144+... =Σ[n=0, ∞] (-1)n ((2n-1)!!/(2n)!!) xn (-1<x<1) |
| 1/√(1-x), x=0 | f(x)=1+x/2+3* x2/8+5*x3/16+35*x4/128+(63*x5)/256+(231*x6)/1024 +(429*x7)/2048+(6435*x =Σ[n=0, ∞] ( (2n-1)!!/(2n)!!) xn (-1<x<1) |
| √(1+x), x=0 | f(x)=1+x/2- x2/8+x3/16-5*x4/128+(7*x5)/256-(21*x^6)/1024+(33*x^7)/2048 -(429*x^8)/32768+(715*x =Σ[n=0, ∞] (-1)n (2(2n-1)!!/((n+1)(2n)!!) xn+1 (-1<x<1) |
| √(1-x), x=0 | f(x)=1-x/2- x2/8-x3/16-5*x4/128-(7*x5)/256-(21*x6)/1024-(33*x^7)/2048 -(429*x8)/32768-(715*x9)/65536-(2431*x10)/262144+... |
| √(1+x2), x=0 | f(x)=1+ x2/2-x4/8+x^6/16-(5*x^8)/128+(7*x^10)/256+... |
| √(1-x2), x=0 | f(x)=1- x2/2-x4/8-x^6/16-(5*x^8)/128-(7*x^10)/256+... |
| 1/√(1+x2), x=0 | f(x)=1- x2/2+3*x4/8-(5*x6)/16+(35*x8)/128-(63*x^10)/256+... =Σ[n=0, ∞] (-1)n (2n-1)!!/(2n)!! x2n (-1<x<1) |
| 1/√(1-x2), x=0 | f(x)=1+ x2/2+3*x4/8+(5*x6)/16+(35*x8)/128+(63*x10)/256+... =Σ[n=0, ∞] (2n-1)!!/(2n)!! x2n (-1<x<1) |
| sin(x), x=0 |
f(x)=x-x3/3!+x5/5!-x7/7!+x9/9!+...+(-1)n
x2n+1 /(2n+1)!+... |
| cos(x), x=0 |
f(x)=1- x2/2!+x4/4!-x^6/6!+x^8/8!-x^10/10!+...+(-1)n x2n /(2n)!+... |
| tan(x), x=0 |
f(x)=x+2x3/3!+16x5/5!+272x7/7!+(7936x9/9!)+(353792x11/11!)+... =x+x3/3+2x5/15+17x7/315+(62x9/2835)+(1382x11/155925)+... (-1<x<1) |
| sin-1 (x), x=0 |
f(x)=x+x3/6+(3*x5)/40+(5*x7)/112+(35*x9)/1152+... |
| cos-1 (x), x=0 |
f(x)=π/2-x-x3/6-(3*x5)/40-(5*x7)/112-(35*x9)/1152+... |
| tan-1 (x), x=0 |
f(x)=x-x3/3+x5/5-x7/7+x9/9+... =Σ[n=0, ∞] (-1)n x2n+1 /(2n+1) (-1<x<1) |
| sinh(x)=(ex-e-x)/2 |
f(x)=x+x3/3!+x5/5!+x7/7!+x9/9!+
... (-1<x<1) =Σ[n=0, ∞] x2n+1/(2n+1)! |
| cosh(x)=(ex+e-x)/2 |
f(x)=1+x2/2!+x4/4!+x6/6!+x8/8!+
... (-1<x<1) =Σ[n=0, ∞] x2n /(2n)! |
| tanh(x)=(ex-e-x)/(ex+e-x) |
f(x)=x-2x3/3!+16x5/5!-272x7/7!+7936x9/9!-353792x11/11!+... =x-x3/3+2x5/15-17x7/315+62x9/2835-1382x11/155925+... (-1<x<1) |
| sinh-1 (x) =in(x+√(1+x2)) |
f(x)=x-(1/6)x3+(3/40)x5-(5/112)x7+(35/1152)x9-
... =Σ[n=0, ∞] {(-1)n (2n-1)!!/((2n+1)(2n)!!)} x2n+1 (-1<x<1) |
| sin(x)/x, x=0 |
f(x)=1- x2/6+x4/120-x^6/5040+x^8/362880-x^10/39916800+... |
| sin(sin(x)), x=0 |
f(x)=x-x3/3+x5/10-(8*x7)/315+(13*x9)/2520-(47*x11)/49896 +(15481*x13)/97297200-(15788*x15)/638512875+... |
| sin(cos(x)), x=0 |
f(x)=sin(1)-(cos(1)*x2)/2-(3*sin(1)-cos(1))*x4/24+((15*sin(1)+14*cos(1))*x6)/720 +((42*sin(1)-209*cos(1))*x8)/40320+... |
| cos(cos(x)), x=0 |
f(x)=cos(1)+(sin(1)*x2)/2-(3*cos(1)+sin(1))*x4/24+((15*cos(1)-14*sin(1))*x6)/720 +((42*cos(1)+209*sin(1))*x8)/40320+... |
| cos(sin(x)), x=0 |
f(x)=1- x2/2+5*x4/24-(37*x6)/720+(457*x8)/40320-(389*x10)/172800 +(599*x12)/1520640-(5497741*x14)/87178291200+(39584029*x16)/4184557977600+... |
[1]Colin Maclaurin
[2]http://mathworld.wolfram.com/MaclaurinSeries.html
[3]http://www.iris.dti.ne.jp/~k-ohkura/physics/taylor.html
[4]http://www.ee.t-kougei.ac.jp/tuushin/lecture/math1/htdocs/function/derivative/maclaurin.html#Maclaurin
[5]
ローラン展開(Laurent series expansion)：負のべき乗展開を含むz=z1=a+b i の周りの展開
| 複素関数ｆ（z)をz=z1=a+i b の周りに展開する |
[wxMaxima] taylor(f(z),z,a+b*%_i,5); [Maple] series(f(z),z,a+b*I,5);type(%,'laurent'); |
| 1/(z^2+1)^2, z=i |
taylor(1/(z^2+1)^2,z,%_i,2); f(z)=-1/(4*(z-i)^2)-i/(4*(z-i))+3/16+(%i*(z-i))/8-(5*(z-i)^2)/64+... |
| z/(z^2+1)^2, z=-i |
taylor(z/(z^2+1)^2,z,-%_i,2); f(z)=i/(4*(z+i)^2)+i/16+(z+i)/16-(3*i*(z+i)^2)/64+... |
