数学の基礎と公式-13

数学の基礎と公式

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12. 級数-4(テイラー展開、マクローリン展開)

(1) $\displaystyle{}$

$\displaystyle{e^x=1+x+{{x^2}\over{2!}}+{{x^3}\over{3!}}+{{x^4}\over{4!}}+{{x^5}\over{5!}}+{{x^6}\over{6!}}+{{x^7}\over{7!}}+{{x^8}\over{8!}}+{{x^9
 }\over{9!}}+{{x^{10}}\over{10!}}+\cdots } $

(2) $\displaystyle{\log(x+1)=x-{{x^2}\over{2}}+{{x^3}\over{3}}-{{x^4}\over{4}}+{{x^5}\over{5}}- {{x^6}\over{6}}+{{x^7}\over{7}}-{{x^8}\over{8}}+{{x^9}\over{9}}-{{x
 ^{10}}\over{10}}+\cdots }$

(3) $\displaystyle{\log(x+2)=\log 2+{{x}\over{2}}-{{x^2}\over{8}}+{{x^3}\over{24}}-{{x^4}\over{ 64}}+{{x^5}\over{160}}-{{x^6}\over{384}}+{{x^7}\over{896}}-{{x^8 }\over{2048}}+{{x^9}\over{4608}}-{{x^{10}}\over{10240}}+\cdots }$

(4) $\displaystyle{\log(x^2+1)=x^2-{{x^4}\over{2}}+{{x^6}\over{3}}-{{x^8}\over{4}}+{{x^{10}}\over{ 5}}+\cdots  }$

(5) $\displaystyle{\log(x^2+2)=\log 2+{{x^2}\over{2}}-{{x^4}\over{8}}+{{x^6}\over{24}}-{{x^8 }\over{64}}+{{x^{10}}\over{160}}+\cdots   }$

(6) $\displaystyle{\log(x^2+x+1)=x+{{x^2}\over{2}}-{{2\,x^3}\over{3}}+{{x^4}\over{4}}+{{x^5}\over{5 }}-{{x^6}\over{3}}+{{x^7}\over{7}}+{{x^8}\over{8}}-{{2\,x^9}\over{9 }}+{{x^{10}}\over{10}}+\cdots   }$

(7) $\displaystyle{\log(x+y)=\log y+{{x}\over{y}}-{{x^2}\over{2\,y^2}}+{{x^3}\over{3\,y^3}}-{{x^ 4}\over{4\,y^4}}+{{x^5}\over{5\,y^5}}-{{x^6}\over{6\,y^6}}+{{x^7 }\over{7\,y^7}}-{{x^8}\over{8\,y^8}}+{{x^9}\over{9\,y^9}}-{{x^{10} }\over{10\,y^{10}}}+\cdots  }$

(8) $\displaystyle{\log \left( \dfrac{1+x}{1-x}\right)=2\,x+{{2\,x^3}\over{3}}+{{2\,x^5}\over{5}}+{{2\,x^7}\over{7}}+{{2\, x^9}\over{9}}+\cdots }$

(9) $\displaystyle{\log \left( x+\sqrt{1+x^2}\right)=x-{{x^3}\over{6}}+{{3\,x^5}\over{40}}-{{5\,x^7}\over{112}}+{{35\,x^ 9}\over{1152}}+\cdots  }$

(10) $\displaystyle{\log \left( x+\sqrt{4+x^2}\right)=\log 2+{{x}\over{2}}-{{x^3}\over{48}}+{{3\,x^5}\over{1280}}-{{5\,x^ 7}\over{14336}}+{{35\,x^9}\over{589824}}+\cdots }$

(11) $\displaystyle{\sin x=x-{{x^3}\over{6}}+{{x^5}\over{120}}-{{x^7}\over{5040}}+{{x^9}\over{ 362880}}+\cdots }$

(12) $\displaystyle{\cos x=1-{{x^2}\over{2}}+{{x^4}\over{24}}-{{x^6}\over{720}}+{{x^8}\over{ 40320}}-{{x^{10}}\over{3628800}}+\cdots}$

(13) $\displaystyle{\tan x=x+{{x^3}\over{3}}+{{2\,x^5}\over{15}}+{{17\,x^7}\over{315}}+{{62\,x ^9}\over{2835}}+\cdots}$

(14)

(15) $\displaystyle{\sec x=1+{{x^2}\over{2}}+{{5\,x^4}\over{24}}+{{61\,x^6}\over{720}}+{{277\, x^8}\over{8064}}+{{50521\,x^{10}}\over{3628800}}+\cdots}$

(16) $\displaystyle{\cot x={{1}\over{x}}-{{x}\over{3}}-{{x^3}\over{45}}-{{2\,x^5}\over{945}}- {{x^7}\over{4725}}-{{2\,x^9}\over{93555}}+\cdots }$

(17)

(18)

(19)

(20) $\displaystyle{\log(\sin x)=\log x -{{x^2}\over{6}}-{{x^4}\over{180}}-{{x^6}\over{2835 }}-{{x^8}\over{37800}}-{{x^{10}}\over{467775}}+\cdots}$

(21) $\displaystyle{\log(\cos x)=-{{x^2}\over{2}}-{{x^4}\over{12}}-{{x^6}\over{45}}-{{17\,x^8}\over{ 2520}}-{{31\,x^{10}}\over{14175}}+\cdots}$

(22) $\displaystyle{\log(\tan x)=\log x +{{x^2}\over{3}}+{{7\,x^4}\over{90}}+{{62\,x^6 }\over{2835}}+{{127\,x^8}\over{18900}}+{{146\,x^{10}}\over{66825}} +\cdots}$

(23) $\displaystyle{e^{\sin x}=1+x+{{x^2}\over{2}}-{{x^4}\over{8}}-{{x^5}\over{15}}-{{x^6}\over{ 240}}+{{x^7}\over{90}}+{{31\,x^8}\over{5760}}+{{x^9}\over{5670}}-{{ 2951\,x^{10}}\over{3628800}}+\cdots }$

(24) $\displaystyle{e^{\cos x}=e-{{e\,x^2}\over{2}}+{{e\,x^4}\over{6}}-{{31\,e\,x^6}\over{720}}+{{ 379\,e\,x^8}\over{40320}}-{{1639\,e\,x^{10}}\over{907200}}+\cdots}$

(25) $\displaystyle{e^{\tan x}=1+x+{{x^2}\over{2}}+{{x^3}\over{2}}+{{3\,x^4}\over{8}}+{{37\,x^5 }\over{120}}+{{59\,x^6}\over{240}}+{{137\,x^7}\over{720}}+{{871\,x^8 }\over{5760}}+{{41641\,x^9}\over{362880}}+{{325249\,x^{10}}\over{
 3628800}}+\cdots }$

(26)

(27)

(28)

(29) $\displaystyle{\rm{arcsinh} x=x-{{x^3}\over{6}}+{{3\,x^5}\over{40}}-{{5\,x^7}\over{112}}+{{35\,x^ 9}\over{1152}}+\cdots }$

(30) $\displaystyle{\rm{arccosh} x=-{{i\,\pi}\over{2}}+i\,x+{{i\,x^3}\over{6}}+{{3\,i\,x^5}\over{40}}+ {{5\,i\,x^7}\over{112}}+{{35\,i\,x^9}\over{1152}}+\cdots }$

(31) $\displaystyle{\rm{arctanh} x=x+{{x^3}\over{3}}+{{x^5}\over{5}}+{{x^7}\over{7}}+{{x^9}\over{9}} +\cdots  }$

(32) $\displaystyle{\rm{arccsch} x=-\log x+\log 2+\cdots +{{x^2}\over{4}}-{{3\,x^4}\over{32}}+{{5\,x^6 }\over{96}}-{{35\,x^8}\over{1024}}+{{63\,x^{10}}\over{2560}}+\cdots  }$

(33) $\displaystyle{\rm{arcsech} x=-\log x+\log 2+\cdots -{{x^2}\over{4}}-{{3\,x^4}\over{32}}-{{5\,x^6 }\over{96}}-{{35\,x^8}\over{1024}}-{{63\,x^{10}}\over{2560}}+\cdots  }$

(33) $\displaystyle{\rm{arccoth} x=-{{\pi\,i}\over{2}}+x+{{x^3}\over{3}}+{{x^5}\over{5}}+{{x^7}\over{7 }}+{{x^9}\over{9}}+\cdots   }$