ローラン級数（Laurent Series）の概念が難しく手を焼いています
ローラン級数 - Wikipedia

負冪の項も含む形での冪級数としての関数の表示のことである。テイラー級数展開できない複素関数を表示する場合に利用される。ローラン級数の名は、最初の発表が1843年にピエール・アルフォンス・ローランによってなされたことに由来する。ローラン級数の概念自体はそれより先の1841年にカール・ワイエルシュトラスによって発見されていたが公表されなかった。

こういう時にはとりあえず例題を動かしてみるに限ります。

関数exp(x)/x+exp(-x)のローラン変換

Macrolin_expansion_Laurent<-function(x){

#グラフのスケール決定
Graph_scale_x<-c(0,3)
Graph_scale_y<-c(0,100)
#関数と凡例の決定
switch(x,
"0"= f0<-function(x) {2*x/x},
"1"= f0<-function(x) {2*(x^-1)+2+x/2},
"2"= f0<-function(x) {(x^-2)/2+2*(x^-1)+2+x/2+(x^2)/6},
"3"= f0<-function(x) {(x^-3)/6+(x^-2)/2+2*(x^-1)+2+x/2+(x^2)/6+(x^3)/24},
"4"= f0<-function(x) {(x^-4)/24+(x^-3)/6+(x^-2)/2+2*(x^-1)+2+x/2+(x^2)/6+(x^3)/24+(x^4)/120},
"5"= f0<-function(x) {(x^-5)/120+(x^-4)/24+(x^-3)/6+(x^-2)/2+2*(x^-1)+2+x/2+(x^2)/6+(x^3)/24+(x^4)/120+(x^5)/720},
"6"= f0<-function(x) {(x^-6)/720+(x^-5)/120+(x^-4)/24+(x^-3)/6+(x^-2)/2+2*(x^-1)+2+x/2+(x^2)/6+(x^3)/24+(x^4)/120+(x^5)/720+(x^6)/5040},
"7"= f0<-function(x) {(x^-7)/5040+(x^-6)/720+(x^-5)/120+(x^-4)/24+(x^-3)/6+(x^-2)/2+2*(x^-1)+2+x/2+(x^2)/6+(x^3)/24+(x^4)/120+(x^5)/720+(x^6)/5040+(x^7)/40320},
"8"= f0<-function(x) {(x^-8)/40320+(x^-7)/5040+(x^-6)/720+(x^-5)/120+(x^-4)/24+(x^-3)/6+(x^-2)/2+2*(x^-1)+2+x/2+(x^2)/6+(x^3)/24+(x^4)/120+(x^5)/720+(x^6)/5040+(x^7)/40320+(x^8)/362880},
"9"= f0<-function(x) {(x^-9)/362880+(x^-8)/40320+(x^-7)/5040+(x^-6)/720+(x^-5)/120+(x^-4)/24+(x^-3)/6+(x^-2)/2+2*(x^-1)+2+x/2+(x^2)/6+(x^3)/24+(x^4)/120+(x^5)/720+(x^6)/5040+(x^7)/40320+(x^8)/362880+(x^9)/3628800}
)

switch(x,
"0"=text<-c("2"),
"1"=text<-c("2x^-1+2+1/2*x/2"),
"2"=text<-c("x^-2/2!+2x^-1+2+1/2*x/2!+x^2/3!"),
"3"=text<-c("x^-3/3!+x^-2/2!+2x^-1+2+1/2*x/2!+x^2/3!+x^3/4!"),
"4"=text<-c("x^-4/4!+x^-3/3!+x^-2/2!+2x^-1+2+1/2*x/2!+x^2/3!+x^3/4!+x^4/5!"),
"5"=text<-c("x^-5/5!+x^-4/4!+x^-3/3!+x^-2/2!+2x^-1+2+1/2*x/2!+x^2/3!+x^3/4!+x^4/5!+x^5/6!"),
"6"=text<-c("x^-6/6!+x^-5/5!+x^-4/4!+x^-3/3!+x^-2/2!+2x^-1+2+1/2*x/2!+x^2/3!+x^3/4!+x^4/5!+x^5/6!+x^6/7!"),
"7"=text<-c("x^-7/7!+x^-6/6!+x^-5/5!+x^-4/4!+x^-3/3!+x^-2/2!+2x^-1+2+1/2*x/2!+x^2/3!+x^3/4!+x^4/5!+x^5/6!+x^6/7!+x^7/8!"),
"8"=text<-c("x^-8/8!+x^-7/7!+x^-6/6!+x^-5/5!+x^-4/4!+x^-3/3!+x^-2/2!+2x^-1+2+1/2*x/2!+x^2/3!+x^3/4!+x^4/5!+x^5/6!+x^6/7!+x^7/8!+x^8/9!"),
"9"=text<-c("x^-9/9!+x^-8/8!+x^-7/7!+x^-6/6!+x^-5/5!+x^-4/4!+x^-3/3!+x^-2/2!+2x^-1+2+1/2*x/2!+x^2/3!+x^3/4!+x^4/5!+x^5/6!+x^6/7!+x^7/8!+x^8/9!+x^9/10!")
)

f1=function(x){exp(x)/x+exp(-x)}

plot(f1,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(0,0,0),main="Laurent expansion", xlab="X", ylab="Y")
par(new=T)#上書き指定
plot(f0,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(1,0,0),main="", xlab="", ylab="")

legend("topright", legend=c("exp(x)/x+exp(-x)",text),lty=c(1,1),col=c(rgb(0,0,0),rgb(1,0,0)))
}

library("animation")
Time_Code=c("0","0","0","1","1","1","2","2","2","3","3","3","4","4","4","5","5","5","6","6","6","7","7","7","8","8","8","9","9","9")
saveGIF({
for (i in Time_Code){
　　Macrolin_expansion_Laurent(i)
}
}, interval = 0.1, movie.name = "Laurent01.gif")

あれ、これで合ってるの？　そもそもテイラー展開への理解が決定的に不足しているので何もわかりません。何しろこれまでは著名な関数のそれの表示に使ってただけですからね。
【Rで球面幾何学】オイラーの公式を導出したマクローリン級数の限界？

#「収束半径が有限」の場合のテイラー展開

そもそもテーラー展開が生成する演算結果集合は、それ自体が演算対象となるので、当然演算規則があります。
第9回テイラー展開の計算／ローラン級数

展開前の関数f(z)について変数をz→wと置き換えてからテイラー展開したものは、f(z)をテイラー展開してからz→wと置き換えたものと同じである。

関数の和f(z)+g(z)の級数展開は、収束半径内では級数の各項の和で与えられる。

関数の積f(z)g(z)の級数展開は、一般にはコーシー積で与えられるが、場合によってはより単純に計算できる場合もある。
コーシー積 - Wikipedia

関数の微分f′(z)、積分 ∫f(z)dzをテイラー展開したものは、収束半径の内部においては関数をテイラー展開してから微分・積分を取ったものと等しい。また、収束半径は微分・積分の前後で同じ値となる。工夫をすれば、この性質を利用してテイラー展開を簡単化することもできる。

そもそも基本中の基本を試してませんでした。

1/(1-x)のテイラー展開

Macrolin_expansion_Laurent<-function(x){

#グラフのスケール決定
Graph_scale_x<-c(0,3)
Graph_scale_y<-c(0,100)
#関数と凡例の決定
switch(x,
"0"= f0<-function(x) {x/x},
"1"= f0<-function(x) {x+1},
"2"= f0<-function(x) {1+x+x^2},
"3"= f0<-function(x) {1+x+x^2+x^3},
"4"= f0<-function(x) {1+x+x^2+x^3+x^4},
"5"= f0<-function(x) {1+x+x^2+x^3+x^4+x^5},
"6"= f0<-function(x) {1+x+x^2+x^3+x^4+x^5+x^6},
"7"= f0<-function(x) {1+x+x^2+x^3+x^4+x^5+x^6+x^7},
"8"= f0<-function(x) {1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8},
"9"= f0<-function(x) {1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9}
)

switch(x,
"0"=text<-c("1"),
"1"=text<-c("1+x"),
"2"=text<-c("1+x+x^2"),
"3"=text<-c("1+x+x^2+x^3"),
"4"=text<-c("1+x+x^2+x^3+x^4"),
"5"=text<-c("1+x+x^2+x^3+x^4+x^5"),
"6"=text<-c("1+x+x^2+x^3+x^4+x^5+x^6"),
"7"=text<-c("1+x+x^2+x^3+x^4+x^5+x^6+x^7"),
"8"=text<-c("1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8"),
"9"=text<-c("1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9")
)

f1=function(x){1/(1-x)}

plot(f1,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(0,0,0),main="Macrolin expansion", xlab="X", ylab="Y")
par(new=T)#上書き指定
plot(f0,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(1,0,0),main="", xlab="", ylab="")

legend("topright", legend=c("1/(1-x)",text),lty=c(1,1),col=c(rgb(0,0,0),rgb(1,0,0)))
}

library("animation")
Time_Code=c("0","0","0","1","1","1","2","2","2","3","3","3","4","4","4","5","5","5","6","6","6","7","7","7","8","8","8","9","9","9")
saveGIF({
for (i in Time_Code){
　　Macrolin_expansion_Laurent(i)
}
}, interval = 0.1, movie.name = "Macrolin02.gif")

1/(1-x)^2のテイラー展開

Macrolin_expansion_Laurent<-function(x){

#グラフのスケール決定
Graph_scale_x<-c(0,3)
Graph_scale_y<-c(0,100)
#関数と凡例の決定
switch(x,
"0"= f0<-function(x) {x/x},
"1"= f0<-function(x) {2*x+1},
"2"= f0<-function(x) {1+2*x+3*x^2},
"3"= f0<-function(x) {1+2*x+3*x^2+4*x^3},
"4"= f0<-function(x) {1+2*x+3*x^2+4*x^3+5*x^4},
"5"= f0<-function(x) {1+2*x+3*x^2+4*x^3+5*x^4+6*x^5},
"6"= f0<-function(x) {1+2*x+3*x^2+4*x^3+5*x^4+6*x^5+7*x^6},
"7"= f0<-function(x) {1+2*x+3*x^2+4*x^3+5*x^4+6*x^5+7*x^6+8*x^7},
"8"= f0<-function(x) {1+2*x+3*x^2+4*x^3+5*x^4+6*x^5+7*x^6+8*x^7+9*x^8},
"9"= f0<-function(x) {1+2*x+3*x^2+4*x^3+5*x^4+6*x^5+7*x^6+8*x^7+9*x^8+10*x^9}
)

switch(x,
"0"=text<-c("1"),
"1"=text<-c("1+2x"),
"2"=text<-c("1+2x+3x^2"),
"3"=text<-c("1+2x+3x^2+4x^3"),
"4"=text<-c("1+2x+3x^2+4x^3+5x^4"),
"5"=text<-c("1+2x+3x^2+4x^3+5x^4+6x^5"),
"6"=text<-c("1+2x+3x^2+4x^3+5x^4+6x^5+7x^6"),
"7"=text<-c("1+2x+3x^2+4x^3+5x^4+6x^5+7x^6+8x^7"),
"8"=text<-c("1+2x+3x^2+4x^3+5x^4+6x^5+7x^6+8x^7+9x^8"),
"9"=text<-c("1+2x+3x^2+4x^3+5x^4+6x^5+7x^6+8x^7+9x^8+10x^9")
)

f1=function(x){1/(1-x)^2}

plot(f1,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(0,0,0),main="Macrolin expansion", xlab="X", ylab="Y")
par(new=T)#上書き指定
plot(f0,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(1,0,0),main="", xlab="", ylab="")

legend("topright", legend=c("1/(1-x)^2",text),lty=c(1,1),col=c(rgb(0,0,0),rgb(1,0,0)))
}

library("animation")
Time_Code=c("0","0","0","1","1","1","2","2","2","3","3","3","4","4","4","5","5","5","6","6","6","7","7","7","8","8","8","9","9","9")
saveGIF({
for (i in Time_Code){
　　Macrolin_expansion_Laurent(i)
}
}, interval = 0.1, movie.name = "Macrolin05.gif")

次いで1/(1+x)のテイラー展開。

Macrolin_expansion_Laurent<-function(x){

#グラフのスケール決定
Graph_scale_x<-c(0,100)
Graph_scale_y<-c(0,2)
#関数と凡例の決定
switch(x,
"0"= f0<-function(x) {x/x},
"1"= f0<-function(x) {1-x},
"2"= f0<-function(x) {1-x+x^2},
"3"= f0<-function(x) {1-x+x^2-x^3},
"4"= f0<-function(x) {1-x+x^2-x^3+x^4},
"5"= f0<-function(x) {1-x+x^2-x^3+x^4-x^5},
"6"= f0<-function(x) {1-x+x^2-x^3+x^4-x^5+x^6},
"7"= f0<-function(x) {1-x+x^2-x^3+x^4-x^5+x^6-x^7},
"8"= f0<-function(x) {1-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8},
"9"= f0<-function(x) {1-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8-x^9}
)

switch(x,
"0"=text<-c("1"),
"1"=text<-c("1-x"),
"2"=text<-c("1-x+x^2"),
"3"=text<-c("1-x+x^2-x^3"),
"4"=text<-c("1-x+x^2-x^3+x^4"),
"5"=text<-c("1-x+x^2-x^3+x^4-x^5"),
"6"=text<-c("1-x+x^2-x^3+x^4-x^5+x^6"),
"7"=text<-c("1-x+x^2-x^3+x^4-x^5+x^6-x^7"),
"8"=text<-c("1-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8"),
"9"=text<-c("1-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8-x^9")
)

f1=function(x){1/(1+x)}

plot(f1,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(0,0,0),main="Macrolin expansion", xlab="X", ylab="Y")
par(new=T)#上書き指定
plot(f0,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(1,0,0),main="", xlab="", ylab="")

legend("topright", legend=c("1/(1+x)",text),lty=c(1,1),col=c(rgb(0,0,0),rgb(1,0,0)))
}

library("animation")
Time_Code=c("0","0","0","1","1","1","2","2","2","3","3","3","4","4","4","5","5","5","6","6","6","7","7","7","8","8","8","9","9","9")
saveGIF({
for (i in Time_Code){
　　Macrolin_expansion_Laurent(i)
}
}, interval = 0.1, movie.name = "Macrolin03.gif")

そのアップ。収束遅い!!

Macrolin_expansion_Laurent<-function(x){

#グラフのスケール決定
Graph_scale_x<-c(0,2)
Graph_scale_y<-c(0,2)
#関数と凡例の決定
switch(x,
"0"= f0<-function(x) {x/x},
"1"= f0<-function(x) {1-x},
"2"= f0<-function(x) {1-x+x^2},
"3"= f0<-function(x) {1-x+x^2-x^3},
"4"= f0<-function(x) {1-x+x^2-x^3+x^4},
"5"= f0<-function(x) {1-x+x^2-x^3+x^4-x^5},
"6"= f0<-function(x) {1-x+x^2-x^3+x^4-x^5+x^6},
"7"= f0<-function(x) {1-x+x^2-x^3+x^4-x^5+x^6-x^7},
"8"= f0<-function(x) {1-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8},
"9"= f0<-function(x) {1-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8-x^9}
)

switch(x,
"0"=text<-c("1"),
"1"=text<-c("1-x"),
"2"=text<-c("1-x+x^2"),
"3"=text<-c("1-x+x^2-x^3"),
"4"=text<-c("1-x+x^2-x^3+x^4"),
"5"=text<-c("1-x+x^2-x^3+x^4-x^5"),
"6"=text<-c("1-x+x^2-x^3+x^4-x^5+x^6"),
"7"=text<-c("1-x+x^2-x^3+x^4-x^5+x^6-x^7"),
"8"=text<-c("1-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8"),
"9"=text<-c("1-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8-x^9")
)

f1=function(x){1/(1+x)}

plot(f1,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(0,0,0),main="Macrolin expansion", xlab="X", ylab="Y")
par(new=T)#上書き指定
plot(f0,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(1,0,0),main="", xlab="", ylab="")

legend("topright", legend=c("1/(1+x)",text),lty=c(1,1),col=c(rgb(0,0,0),rgb(1,0,0)))
}

library("animation")
Time_Code=c("0","0","0","1","1","1","2","2","2","3","3","3","4","4","4","5","5","5","6","6","6","7","7","7","8","8","8","9","9","9")
saveGIF({
for (i in Time_Code){
　　Macrolin_expansion_Laurent(i)
}
}, interval = 0.1, movie.name = "Macrolin04.gif")

ついでに1/(1+x)^2のテイラー展開。

Macrolin_expansion_Laurent<-function(x){

#グラフのスケール決定
Graph_scale_x<-c(0,100)
Graph_scale_y<-c(0,2)
#関数と凡例の決定
switch(x,
"0"= f0<-function(x) {x/x},
"1"= f0<-function(x) {1-2*x},
"2"= f0<-function(x) {1-2*x+3*x^2},
"3"= f0<-function(x) {1-2*x+3*x^2-4*x^3},
"4"= f0<-function(x) {1-2*x+3*x^2-4*x^3+5*x^4},
"5"= f0<-function(x) {1-2*x+3*x^2-4*x^3+5*x^4-6*x^5},
"6"= f0<-function(x) {1-2*x+3*x^2-4*x^3+5*x^4-6*x^5+7*x^6},
"7"= f0<-function(x) {1-2*x+3*x^2-4*x^3+5*x^4-6*x^5+7*x^6-8*x^7},
"8"= f0<-function(x) {1-2*x+3*x^2-4*x^3+5*x^4-6*x^5+7*x^6-8*x^7+9*x^8},
"9"= f0<-function(x) {1-2*x+3*x^2-4*x^3+5*x^4-6*x^5+7*x^6-8*x^7+9*x^8-10*x^9}
)

switch(x,
"0"=text<-c("1"),
"1"=text<-c("1-2x"),
"2"=text<-c("1-2x+3x^2"),
"3"=text<-c("1-2x+3x^2-4x^3"),
"4"=text<-c("1-2x+3x^2-4x^3+5x^4"),
"5"=text<-c("1-2x+3x^2-4x^3+5x^4-6x^5"),
"6"=text<-c("1-2x+3x^2-4x^3+5x^4-6x^5+7x^6"),
"7"=text<-c("1-2x+3x^2-4x^3+5x^4-6x^5+7x^6-8x^7"),
"8"=text<-c("1-2x+3x^2-4x^3+5x^4-6x^5+7x^6-8x^7+9x^8"),
"9"=text<-c("1-2x+3x^2-4x^3+5x^4-6x^5+7x^6-8x^7+9x^8-10x^9")
)

f1=function(x){1/(1+x)^2}

plot(f1,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(0,0,0),main="Macrolin expansion", xlab="X", ylab="Y")
par(new=T)#上書き指定
plot(f0,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(1,0,0),main="", xlab="", ylab="")

legend("topright", legend=c("1/(1+x)^2",text),lty=c(1,1),col=c(rgb(0,0,0),rgb(1,0,0)))
}

library("animation")
Time_Code=c("0","0","0","1","1","1","2","2","2","3","3","3","4","4","4","5","5","5","6","6","6","7","7","7","8","8","8","9","9","9")
saveGIF({
for (i in Time_Code){
　　Macrolin_expansion_Laurent(i)
}
}, interval = 0.1, movie.name = "Macrolin06.gif")

そのアップ。

Macrolin_expansion_Laurent<-function(x){

#グラフのスケール決定
Graph_scale_x<-c(0,2)
Graph_scale_y<-c(0,2)
#関数と凡例の決定
switch(x,
"0"= f0<-function(x) {x/x},
"1"= f0<-function(x) {1-2*x},
"2"= f0<-function(x) {1-2*x+3*x^2},
"3"= f0<-function(x) {1-2*x+3*x^2-4*x^3},
"4"= f0<-function(x) {1-2*x+3*x^2-4*x^3+5*x^4},
"5"= f0<-function(x) {1-2*x+3*x^2-4*x^3+5*x^4-6*x^5},
"6"= f0<-function(x) {1-2*x+3*x^2-4*x^3+5*x^4-6*x^5+7*x^6},
"7"= f0<-function(x) {1-2*x+3*x^2-4*x^3+5*x^4-6*x^5+7*x^6-8*x^7},
"8"= f0<-function(x) {1-2*x+3*x^2-4*x^3+5*x^4-6*x^5+7*x^6-8*x^7+9*x^8},
"9"= f0<-function(x) {1-2*x+3*x^2-4*x^3+5*x^4-6*x^5+7*x^6-8*x^7+9*x^8-10*x^9}
)

switch(x,
"0"=text<-c("1"),
"1"=text<-c("1-2x"),
"2"=text<-c("1-2x+3x^2"),
"3"=text<-c("1-2x+3x^2-4x^3"),
"4"=text<-c("1-2x+3x^2-4x^3+5x^4"),
"5"=text<-c("1-2x+3x^2-4x^3+5x^4-6x^5"),
"6"=text<-c("1-2x+3x^2-4x^3+5x^4-6x^5+7x^6"),
"7"=text<-c("1-2x+3x^2-4x^3+5x^4-6x^5+7x^6-8x^7"),
"8"=text<-c("1-2x+3x^2-4x^3+5x^4-6x^5+7x^6-8x^7+9x^8"),
"9"=text<-c("1-2x+3x^2-4x^3+5x^4-6x^5+7x^6-8x^7+9x^8-10x^9")
)

f1=function(x){1/(1+x)^2}

plot(f1,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(0,0,0),main="Macrolin expansion", xlab="X", ylab="Y")
par(new=T)#上書き指定
plot(f0,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(1,0,0),main="", xlab="", ylab="")

legend("topright", legend=c("1/(1+x)^2",text),lty=c(1,1),col=c(rgb(0,0,0),rgb(1,0,0)))
}

library("animation")
Time_Code=c("0","0","0","1","1","1","2","2","2","3","3","3","4","4","4","5","5","5","6","6","6","7","7","7","8","8","8","9","9","9")
saveGIF({
for (i in Time_Code){
　　Macrolin_expansion_Laurent(i)
}
}, interval = 0.1, movie.name = "Macrolin07.gif")

とりあえず「関数に収束半径がある」とはこういうイメージで、それに対してテイラー級数/マクローリン級数はこう展開されるという話ですね。

そういえばlog(1+x)と-log(1-x)と両者の和log((1-x)/(1+x))(収束半径1)という特殊ケースについては既に邂逅済みだったが、この場合もまだローラン変換の出番はなかったのである。
【Rで球面幾何学】オイラーの公式を導出したマクローリン級数の限界？

#最も簡単なローラン級数(Laurent Series)の例

第9回テイラー展開の計算／ローラン級数

1/(z(1−z))を原点z0=0の周りでローラン展開するときは、1/z×1/(1-z)と書き換えてから、1/(1−z)を原点z0=0の周りでテイラー展開すればよい。この級数の収束領域は、円形領域|z|<1から**原点**z=0を除いた領域となる。

Laurent_expansion<-function(x){

#グラフのスケール決定
Graph_scale_x<-c(0,3)
Graph_scale_y<-c(0,100)
#関数と凡例の決定
switch(x,
"0"= f0<-function(x) {1/x},
"1"= f0<-function(x) {1/x+1},
"2"= f0<-function(x) {1/x+1+x},
"3"= f0<-function(x) {1/x+1+x+x^2},
"4"= f0<-function(x) {1/x+1+x+x^2+x^3},
"5"= f0<-function(x) {1/x+1+x+x^2+x^3+x^4},
"6"= f0<-function(x) {1/x+1+x+x^2+x^3+x^4+x^5},
"7"= f0<-function(x) {1/x+1+x+x^2+x^3+x^4+x^5+x^6},
"8"= f0<-function(x) {1/x+1+x+x^2+x^3+x^4+x^5+x^6+x^7},
"9"= f0<-function(x) {1/x+1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8}
)

switch(x,
"0"=text<-c("1/x"),
"1"=text<-c("1/x+1"),
"2"=text<-c("1/x+1+x"),
"3"=text<-c("1/x+1+x+x^2"),
"4"=text<-c("1/x+1+x+x^2+x^3"),
"5"=text<-c("1/x+1+x+x^2+x^3+x^4"),
"6"=text<-c("1/x+1+x+x^2+x^3+x^4+x^5"),
"7"=text<-c("1/x+1+x+x^2+x^3+x^4+x^5+x^6"),
"8"=text<-c("1/x+1+x+x^2+x^3+x^4+x^5+x^6+x^7"),
"9"=text<-c("1/x+1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8")
)

f1=function(x){1/(x*(1-x))}

plot(f1,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(0,0,0),main="Laurent expansion", xlab="X", ylab="Y")
par(new=T)#上書き指定
plot(f0,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(1,0,0),main="", xlab="", ylab="")

legend("topright", legend=c("1/(x*(1-x)",text),lty=c(1,1),col=c(rgb(0,0,0),rgb(1,0,0)))
}

library("animation")
Time_Code=c("0","0","0","1","1","1","2","2","2","3","3","3","4","4","4","5","5","5","6","6","6","7","7","7","8","8","8","9","9","9")
saveGIF({
for (i in Time_Code){
　　Laurent_expansion(i)
}
}, interval = 0.1, movie.name = "Laurent02.gif")

同じ関数1/(z(1−z))を原点z0=1の周りでローラン展開するときは、1/(1-z)×(-1/z)と書き換えてから、-1/zを原点z0=1の周りでテイラー展開すればよい。この級数の収束領域は、円形領域|z|<1から**原点**z=1を除いた領域となる。

＊何か間違ってる気もするけど、現時点の私には直せません(2021.2.12)

Laurent_expansion<-function(x){

#グラフのスケール決定
Graph_scale_x<-c(0,3)
Graph_scale_y<-c(-100,100)
#関数と凡例の決定
switch(x,
"0"= f0<-function(x) {-1/(1-x)},
"1"= f0<-function(x) {-1/(1-x)+1},
"2"= f0<-function(x) {-1/(1-x)+1-(x-1)},
"3"= f0<-function(x) {-1/(1-x)+1-(x-1)+(x-1)^2},
"4"= f0<-function(x) {-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3},
"5"= f0<-function(x) {-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3+(x-1)^4},
"6"= f0<-function(x) {-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3+(x-1)^4+(x-1)^5},
"7"= f0<-function(x) {-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3+(x-1)^4+(x-1)^5+(x-1)^6},
"8"= f0<-function(x) {-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3+(x-1)^4+(x-1)^5+(x-1)^6+(x-1)^7},
"9"= f0<-function(x) {-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3+(x-1)^4+(x-1)^5+(x-1)^6+(x-1)^7+(x-1)^8}
)

switch(x,
"0"=text<-c("-1/(1-x)"),
"1"=text<-c("-1/(1-x)+1"),
"2"=text<-c("-1/(1-x)+1-(x-1)"),
"3"=text<-c("-1/(1-x)+1-(x-1)+(x-1)^2"),
"4"=text<-c("-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3"),
"5"=text<-c("-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3+(x-1)^4"),
"6"=text<-c("-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3+(x-1)^4+(x-1)^5"),
"7"=text<-c("-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3+(x-1)^4+(x-1)^5+(x-1)^6"),
"8"=text<-c("-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3+(x-1)^4+(x-1)^5+(x-1)^6+(x-1)^7"),
"9"=text<-c("-1/(1-x)+(x-1)-(x-1)^2+(x-1)^3+(x-1)^4+(x-1)^5+(x-1)^6+(x-1)^7+(x-1)^8")
)

f1=function(x){1/(x*(1-x))}

plot(f1,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(0,0,0),main="Laurent expansion", xlab="X", ylab="Y")
par(new=T)#上書き指定
plot(f0,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(1,0,0),main="", xlab="", ylab="")

legend("topright", legend=c("1/(x*(1-x)",text),lty=c(1,1),col=c(rgb(0,0,0),rgb(1,0,0)))
}

library("animation")
Time_Code=c("0","0","0","1","1","1","2","2","2","3","3","3","4","4","4","5","5","5","6","6","6","7","7","7","8","8","8","9","9","9")
saveGIF({
for (i in Time_Code){
　　Laurent_expansion(i)
}
}, interval = 0.1, movie.name = "Laurent03.gif")

符号を逆転してみたら混沌が広がりました。チェルノブイリ!!(2021.2.12)。

Laurent_expansion<-function(x){

#グラフのスケール決定
Graph_scale_x<-c(0,3)
Graph_scale_y<-c(-100,100)
#関数と凡例の決定
switch(x,
"0"= f0<-function(x) {1/(1+x)},
"1"= f0<-function(x) {1/(1+x)+1},
"2"= f0<-function(x) {1/(1+x)+1-(x+1)},
"3"= f0<-function(x) {1/(1+x)+1-(x+1)+(x+1)^2},
"4"= f0<-function(x) {1/(1+x)+1-(x+1)+(x+1)^2+(x+1)^3},
"5"= f0<-function(x) {1/(1+x)+1-(x+1)+(x+1)^2+(x+1)^3+(x+1)^4},
"6"= f0<-function(x) {1/(1+x)+1-(x+1)+(x+1)^2+(x+1)^3+(x+1)^4+(x+1)^5},
"7"= f0<-function(x) {1/(1+x)+1-(x+1)+(x+1)^2+(x+1)^3+(x+1)^4+(x+1)^5+(x+1)^6},
"8"= f0<-function(x) {1/(1+x)+1-(x+1)+(x+1)^2+(x+1)^3+(x+1)^4+(x+1)^5+(x+1)^6+(x+1)^7},
"9"= f0<-function(x) {1/(1+x)+1-(x+1)+(x+1)^2+(x+1)^3+(x+1)^4+(x+1)^5+(x+1)^6+(x+1)^7+(x+1)^8}
)

switch(x,
"0"=text<-c("-1/(1-x)"),
"1"=text<-c("-1/(1-x)+1"),
"2"=text<-c("-1/(1-x)+1-(x-1)"),
"3"=text<-c("-1/(1-x)+1-(x-1)+(x-1)^2"),
"4"=text<-c("-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3"),
"5"=text<-c("-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3+(x-1)^4"),
"6"=text<-c("-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3+(x-1)^4+(x-1)^5"),
"7"=text<-c("-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3+(x-1)^4+(x-1)^5+(x-1)^6"),
"8"=text<-c("-1/(1-x)+1-(x-1)+(x-1)^2+(x-1)^3+(x-1)^4+(x-1)^5+(x-1)^6+(x-1)^7"),
"9"=text<-c("-1/(1-x)+(x-1)-(x-1)^2+(x-1)^3+(x-1)^4+(x-1)^5+(x-1)^6+(x-1)^7+(x-1)^8")
)

f1=function(x){1/(x*(1-x))}

plot(f1,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(0,0,0),main="Laurent expansion", xlab="X", ylab="Y")
par(new=T)#上書き指定
plot(f0,xlim=Graph_scale_x,ylim=Graph_scale_y,type="l",col=rgb(1,0,0),main="", xlab="", ylab="")

legend("topright", legend=c("1/(x*(1-x)",text),lty=c(1,1),col=c(rgb(0,0,0),rgb(1,0,0)))
}

library("animation")
Time_Code=c("0","0","0","1","1","1","2","2","2","3","3","3","4","4","4","5","5","5","6","6","6","7","7","7","8","8","8","9","9","9")
saveGIF({
for (i in Time_Code){
　　Laurent_expansion(i)
}
}, interval = 0.1, movie.name = "Laurent04.gif")

もっと複素解析についての基礎知識を蓄えないとローラン変換の世界(つまりそれを必要とする領域)には進めない様です。それが分かった時点で以下続報…