{"id":2026,"date":"2019-06-30T02:18:30","date_gmt":"2019-06-29T17:18:30","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=2026"},"modified":"2020-11-22T16:10:00","modified_gmt":"2020-11-22T07:10:00","slug":"calculus-applications-of-taylor-series","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-applications-of-taylor-series\/","title":{"rendered":"\ud14c\uc77c\ub7ec \uae09\uc218\uc758 \ud65c\uc6a9"},"content":{"rendered":"<p><a href=\"\/blog\/articles\/calculus-taylor-series-and-maclaurin-series\/\">\uc9c0\ub09c \ud3ec\uc2a4\ud2b8<\/a>\uc5d0\uc11c \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uc815\uc758\ud558\uace0 \ud568\uc218\ub97c \ud14c\uc77c\ub7ec \uae09\uc218\ub85c \ub098\ud0c0\ub0b4\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \ub610\ud55c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \\(f\\)\uc5d0 \uc218\ub834\ud568\uc744 \uc99d\uba85\ud558\ub294 \ubc29\ubc95\ub3c4 \uc0b4\ud3b4\ubcf4\uc558\ub2e4.<\/p>\n<p>\ub354\ubd88\uc5b4 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uc0bc\uac01\ud568\uc218\ub97c \uc815\uc758\ud558\ub294 \ubc29\ubc95(<a href=\"\/blog\/articles\/calculus-analytic-definition-of-trigonometric-functions\/\">\uad00\ub828 \ud3ec\uc2a4\ud2b8<\/a>)\uacfc \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uc9c0\uc218\ud568\uc218\ub97c \uc815\uc758\ud558\ub294 \ubc29\ubc95(<a href=\"\/blog\/articles\/calculus-analytic-definition-of-exponential-function\/\">\uad00\ub828 \ud3ec\uc2a4\ud2b8<\/a>)\ub3c4 \uc0b4\ud3b4\ubcf4\uc558\ub2e4.<\/p>\n<\/p>\n<p>\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \ud65c\uc6a9\ud55c \ub2e4\uc591\ud55c \uc608\ub97c \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<div class=\"box contentindex\">\n<h3 class=\"indextitle\">\ub0b4\uc6a9 \uc21c\uc11c<\/h3>\n<ul>\n<li><a href=\"#binomialseries\">\uc774\ud56d\uae09\uc218<\/a><\/li>\n<li><a href=\"#nonelementaryintegral\">\ube44\ucd08\ub4f1\uc801\ubd84<\/a><\/li>\n<li><a href=\"#leibnizsformula\">\ub77c\uc774\ud504\ub2c8\uce20 \uacf5\uc2dd<\/a><\/li>\n<li><a href=\"#indeterminatelimit\">\ubd80\uc815\ud615 \uadf9\ud55c\uc758 \uacc4\uc0b0<\/a><\/li>\n<li><a href=\"#eulersidentity\">\ud568\uc218\uc758 \uc815\uc758\uc5ed \ud655\uc7a5\ud558\uae30<\/a><\/li>\n<\/ul>\n<h3 class=\"pretitle\">\ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9<\/h3>\n<ul>\n<li>\uac70\ub4ed\uc81c\uacf1\uae09\uc218 (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Power_series\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\ud14c\uc77c\ub7ec \uae09\uc218 (<a href=\"\/blog\/articles\/calculus-taylor-series-and-maclaurin-series\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\uc774\ud56d\uc815\ub9ac (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Binomial_theorem\">\uad00\ub828 \uae00<\/a>)<\/li>\n<li>\ub85c\ud53c\ud0c8\uc758 \ubc95\uce59 (<a href=\"\/blog\/articles\/calculus-indeterminate-forms-and-lhopitals-rule\/\">\uad00\ub828 \uae00<\/a>)<\/li>\n<\/ul>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"binomialseries\"><\/a><\/p>\n<h3>\uc774\ud56d\uae09\uc218<\/h3>\n<p>\uc774\ud56d\uc815\ub9ac\uc5d0 \uc758\ud558\uba74 \\(n\\)\uc774 \\(2\\) \uc774\uc0c1\uc778 \uc790\uc5f0\uc218\uc77c \ub54c<br \/>\n\\[(1+x)^n = {}_n \\mathrm{C} _0 + {}_n \\mathrm{C} _1 x^1 + {}_n \\mathrm{C} _2 x^2 + {}_n \\mathrm{C} _3 x^3 + \\cdots + {}_n \\mathrm{C} _n x^n\\]<br \/>\n\uc774\ub2e4. \uc608\ucee8\ub300 \\(n=4\\)\uc774\uba74<br \/>\n\\[\\begin{align}<br \/>\n(1+x)^4 &#038;= {}_4 \\mathrm{C} _0 + {}_4 \\mathrm{C} _1 x + {}_4 \\mathrm{C} _2 x^2 + {}_4 \\mathrm{C} _3 x^3 + {}_4 \\mathrm{C} _4 x^4\\\\[8pt]<br \/>\n&#038;= 1 + 4x + 6x^2 + 4x^3 + x^4<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(_4 \\mathrm{C} _3\\)\uc744 \uacc4\uc0b0\ud558\ub294 \uacfc\uc815\uc744 \ubcf4\uba74<br \/>\n\\[_4 \\mathrm{C} _3 = \\frac{4\\times 3 \\times 2}{3!}\\]<br \/>\n\uc774\ub2e4. \ubd84\uc790\ub294 \\(4\\)\ubd80\ud130 \uc2dc\uc791\ud558\uc5ec \\(1\\)\uc529 \uc904\uc5ec\uac00\uba70 \\(3\\)\uac1c\ub97c \uacf1\ud558\uace0, \ubd84\ubaa8\ub294 \\(3!\\)\uc774\ub2e4. \uc774\uac83\uc744 \uc77c\ubc18\ud654\ud558\uba74 \\(m\\)\uc774 \uc2e4\uc218\uc774\uace0 \\(k\\)\uac00 \uc790\uc5f0\uc218\uc77c \ub54c<br \/>\n\\[_m \\mathrm{C} _k = \\frac{m(m-1)(m-2)(m-3) \\cdots (m-k+1)}{k!}\\]<br \/>\n\uc73c\ub85c \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4. \ud2b9\ud788 \\(m\\)\uc774 \uc815\uc218\uac00 \uc544\ub2c8\uc5b4\ub3c4 \uc774\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \uc77c\ubc18\ud654\ub41c \uc774\ud56d\uacc4\uc218\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"definition\" style=\"padding-bottom: 0em\">\n<p><span class=\"definition\">\uc815\uc758 1. (\uc774\ud56d\uacc4\uc218)<\/span><\/p>\n<p>\\(m\\)\uc774 \uc2e4\uc218\uc774\uace0 \\(k\\)\uac00 \uc790\uc5f0\uc218\uc77c \ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{gather}<br \/>\n\\binom{m}{0} = 1 ,\\quad \\quad \\binom{m}{1} = m ,\\\\[12pt]<br \/>\n\\binom{m}{k} = \\frac{m(m-1)(m-2)(m-3) \\cdots (m-k+1)}{k!} .<br \/>\n\\end{gather}\\]\n<\/p>\n<\/div>\n<p>\uc608\ucee8\ub300 \\(m=-1\\)\uc77c \ub54c<br \/>\n\\[\\begin{gather}<br \/>\n\\binom{-1}{1} = -1 ,\\quad \\binom{-1}{2} = \\frac{-1 \\times (-2)}{2!} = 1,\\\\[12pt]<br \/>\n\\binom{-1}{k} = \\frac{-1(-2)(-3) \\cdots (-1-k+1)}{k!} = (-1)^k \\frac{k!}{k!} = (-1)^k<br \/>\n\\end{gather}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p>\ub2e4\ub978 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<br \/>\n\\[\\begin{align}<br \/>\n\\binom{4}{0} &#038;= 1,\\\\[4pt]<br \/>\n\\binom{4}{1} &#038;= \\frac{4}{1!} = 4, \\\\[4pt]<br \/>\n\\binom{4}{2} &#038;= \\frac{4 \\times 3 }{2!} = 6 , \\\\[4pt]<br \/>\n\\binom{4}{3} &#038;= \\frac{4 \\times 3 \\times 2}{3!} = 4 , \\\\[4pt]<br \/>\n\\binom{4}{4} &#038;= \\frac{4 \\times 3 \\times 2 \\times 1}{4!} = 1,\\\\[4pt]<br \/>\n\\binom{4}{5} &#038;= \\frac{4 \\times 3 \\times 2 \\times 1 \\times 0}{5!} = 0 , \\\\[4pt]<br \/>\n\\binom{4}{k} &#038;= 0 \\quad \\quad (k > 4)<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[(1+x)^4 = 1+ \\sum_{k=1}^{\\infty} \\binom{4}{k} x^k\\]<br \/>\n\uc774\ub2e4. \uadf8\ub807\ub2e4\uba74 \uc9c0\uc218\uac00 \uc790\uc5f0\uc218\uac00 \uc544\ub2cc \uacbd\uc6b0\uc5d0\ub3c4 \uc774\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc744\uae4c? \uc989<br \/>\n\\[(1+x)^m = 1+ \\sum_{k=1}^{\\infty} \\binom{m}{k} x^k \\tag{1.1}\\]<br \/>\n\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\uc744\uae4c? \\(f(x) = (1+x)^m\\)\uc774\ub77c\uace0 \ud558\uace0 \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uad6c\ud574\ubcf4\uc790.<br \/>\n\\[\\begin{align}<br \/>\nf(x) &#038;= (1+x)^m \\\\[6pt]<br \/>\nf &#8216; (x) &#038;= m(1+x)^{m-1} \\\\[6pt]<br \/>\nf &#8216; &#8216; (x) &#038;= m(m-1) (1+x)^{m-2} \\\\[6pt]<br \/>\nf ^{(3)} (x) &#038;= m(m-1)(m-2) (1+x)^{m-3} \\\\[6pt]<br \/>\n&#038;\\,\\,\\,\\vdots \\\\[6pt]<br \/>\nf^{(k)}(x) &#038;= m(m-1)(m-2) \\cdots (m-k+1) (1+x)^{m-k}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\ub294<br \/>\n\\[\\begin{align}<br \/>\n1 + mx + \\frac{m(m-1)}{2!} x^2 &#038;+ \\frac{m(m-1)(m-2)}{3!} x^3 + \\cdots \\\\[12pt]<br \/>\n&#038;+ \\frac{m(m-1)(m-2) \\cdots (m-k+1)}{k!} x^k + \\cdots \\end{align}\\]<br \/>\n\uc774\ub2e4. \uc774 \uc2dd\uc740 (1.1)\uc758 \uc6b0\ubcc0\uacfc \uc77c\uce58\ud55c\ub2e4. \ub610\ud55c<br \/>\n\\[u_{k} = \\binom{m}{k}\\]<br \/>\n\ub77c\uace0 \ud558\uba74<br \/>\n\\[\\lim_{k\\to\\infty} \\left\\lvert \\frac{u_{k+1}}{u_k} \\right\\rvert = \\lim_{k\\to\\infty} \\left\\lvert \\frac{m-k}{k+1}  \\right\\rvert = 1\\]<br \/>\n\uc774\ubbc0\ub85c (1.1)\uc758 \uc6b0\ubcc0\uc758 \uc218\ub834\ubc18\uacbd\uc740 \\(1\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.\n<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 2. (\uc774\ud56d\uae09\uc218)<\/span><\/p>\n<p>\\((1+x)^m\\)\uc758 \uc774\ud56d\uae09\uc218\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[(1+x)^m \\,\\sim\\, 1+ \\sum_{k=1}^{\\infty} \\binom{m}{k} x^k . \\tag{1.2}\\]\n<\/p>\n<\/div>\n<p>\uc704 \uc815\uc758\uc758 \uc2dd (1.2)\uc5d0\uc11c \ub4f1\ud638\ub97c \uc0ac\uc6a9\ud558\uc9c0 \uc54a\uace0 \ubb3c\uacb0 \ud45c\uc2dc\ub97c \uc0ac\uc6a9\ud55c \uac83\uc740 (1.2)\uc758 \uc6b0\ubcc0\uc774 \uc88c\ubcc0\uc5d0 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc9c0 \uc54a\uc558\uae30 \ub54c\ubb38\uc774\ub2e4. \uc774\uc81c \\( \\lvert x \\rvert < 1\\)\uc758 \ubc94\uc704\uc5d0\uc11c (1.2)\uc758 \uc6b0\ubcc0\uc774 \\((1+x)^m\\)\uc5d0 \uc218\ub834\ud568\uc744 \uc99d\uba85\ud574 \ubcf4\uc790. \\(m=0\\)\uc77c \ub54c\ub294 \uc790\uba85\ud558\ubbc0\ub85c \\(m\\ne 0\\)\uc77c \ub54c\ub9cc \uc99d\uba85\ud558\uba74 \ub41c\ub2e4.\n\\[f(x) = 1 + mx + \\frac{m(m-1)}{2!}x^2 + \\frac{m(m-1)(m-2)}{3!} x^3 + \\cdots\\tag{1.3}\\]\n\uc774\ub77c\uace0 \ud558\uc790. \uc774 \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74\n\\[\\begin{align}\nf ' (x) \n&#038;= m + \\frac{m(m-1)}{1} x + \\frac{m(m-1)(m-2)}{2!} x^2 + \\cdots \\\\[3pt]\n&#038;= m\\left\\{ 1 + (m-1)x + \\frac{(m-1)(m-2)}{2!} x^2 + \\cdots \\right\\} \\\\[3pt]\n&#038;= \\frac{m}{1+x} \\left\\{ 1 + (m-1)x + \\frac{(m-1)(m-2)}{2!} x^2 + \\cdots \\right . \\\\[3pt]\n&#038; \\quad\\quad\\quad\\quad\\quad\\quad + \\left . x\\left( 1 + (m-1)x + \\frac{(m-1)(m-2)}{2!} x^2 + \\cdots \\right) \\right\\} \\\\[3pt]\n&#038;= \\frac{m}{1+x} \\left\\{ 1 + (m-1)x + \\frac{(m-1)(m-2)}{2!} x^2 + \\cdots \\right . \\\\[3pt]\n&#038; \\quad\\quad\\quad\\quad\\quad\\quad + \\left . x + (m-1)x^2 + \\frac{(m-1)(m-2)}{2!} x^3 + \\cdots \\right\\} \\\\[3pt]\n&#038;= \\frac{m}{1+x} \\left\\{ 1 + mx + \\frac{m(m-1)}{2!} x^2 + \\cdots \\right\\} \n= \\frac{m}{1+x} f(x) \n\\end{align}\\]\n\uc774\ubbc0\ub85c\n\\[f ' (x) = \\frac{m f(x)}{1+x}\\tag{1.4}\\]\n\uc774\ub2e4.\n\\[g(x) = \\frac{f(x)}{(1+x)^m}\\tag{1.5}\\]\n\ub77c\uace0 \ud558\uace0 (1.4)\ub97c \uc774\uc6a9\ud558\uc5ec \\(g\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74\n\\[\\begin{align}\ng ' (x) &#038;= \\frac{f ' (x) (1+x)^m - mf(x) (1+x)^{m-1}}{(1+x)^{2m}} \\\\[3pt]\n&#038;= \\frac{mf(x) (1+x)^{m-1} - mf(x) (1+x)^{m-1}}{(1+x)^{2m}} =0\n\\end{align}\\]\n\uc774\ubbc0\ub85c \\(g\\)\ub294 \uc0c1\uc218\ud568\uc218\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(g(0)=1\\)\uc774\ubbc0\ub85c \\(\\lvert x \\rvert < 1\\)\uc77c \ub54c\n\\[\\frac{f(x)}{(1+x)^m} = 1 \\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\lvert x \\rvert < 1\\)\uc77c \ub54c\n\\[f(x) = (1+x)^m\\tag{1.6}\\]\n\uc774\ub2e4. \uc989 \\(\\lvert x \\rvert < 1\\)\uc77c \ub54c (1.2)\uc758 \ubb3c\uacb0 \uae30\ud638\ub97c \ub4f1\ud638\ub85c \ubc14\uafbc \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span><\/p>\n<p>\\(f(x) = \\sqrt{1+x}\\)\ub77c\uace0 \ud558\uc790. \\(\\lvert x \\rvert\\)\uac00 \\(0\\)\uc5d0 \uac00\uae4c\uc6b8 \ub54c \\(f(x)\\)\uc758 \uc77c\ucc28\uadfc\uc0ac\uc2dd\uc740<br \/>\n\\[f(x) \\,\\approx\\, 1 + \\frac{x}{2}\\]<br \/>\n\uc774\ub2e4. \uc774\ud56d\uae09\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \ub354 \ub192\uc740 \ucc28\uc218\uc758 \uadfc\uc0ac\uc2dd\uc744 \uad6c\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nf(x) &#038;= (1+x)^{1\/2} \\\\[4pt]<br \/>\n&#038;= 1 + \\frac{x}{2} + \\frac{0.5 \\times(-0.5)}{2!} x^2 + \\frac{0.5 \\times (-0.5) \\times (-1.5)}{3!} x^3 + \\cdots<br \/>\n\\end{align}\\]<br \/>\n\uc774\ubbc0\ub85c \\(\\lvert x \\rvert\\)\uc758 \uac12\uc774 \\(0\\)\uc5d0 \uac00\uae4c\uc6b8 \ub54c<br \/>\n\\[(1+x)^{1\/2} \\,\\approx\\, 1 + \\frac{x}{2} &#8211; \\frac{x^2}{8}\\tag{1.7}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \ub610\ud55c (1.7)\uc5d0\uc11c \\(x\\)\ub97c \ub2e4\ub978 \uc2dd\uc73c\ub85c \ubc14\uafb8\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uadfc\uc0ac\uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{gather}<br \/>\n\\sqrt{1-x^2} \\,\\approx\\, 1 &#8211; \\frac{x^2}{2} &#8211; \\frac{x^4}{8} \\quad\\quad \\lvert x \\rvert \\ll 1 ,\\\\[3pt]<br \/>\n\\sqrt{1-\\frac{1}{x}} \\,\\approx\\, 1 &#8211; \\frac{1}{2x} &#8211; \\frac{1}{8x^2} \\quad\\quad \\lvert x \\rvert \\gg 1 .<br \/>\n\\end{gather}\\]\n<\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"nonelementaryintegral\"><\/a><\/p>\n<h3>\ube44\ucd08\ub4f1\uc801\ubd84<\/h3>\n<p>\ub2e4\uc74c \uc801\ubd84\uc740 \ube5b\uc758 \ud68c\uc808 \ud604\uc0c1\uc744 \uc124\uba85\ud560 \ub54c \ub4f1\uc7a5\ud55c\ub2e4.<br \/>\n\\[\\int \\sin x^2\\,dx \\tag{2.1}\\]<br \/>\n\uadf8\ub7ec\ub098 \uc774 \ubd80\uc815\uc801\ubd84\uc740 \uc720\ub9ac\ud568\uc218, \ubb34\ub9ac\ud568\uc218, \uc0bc\uac01\ud568\uc218, \uc9c0\uc218\ud568\uc218, \ub85c\uadf8\ud568\uc218\uc758 \uacb0\ud569\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc5c6\ub2e4. \ub300\uc2e0 \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uc774 \ubd80\uc815\uc801\ubd84\uc744 \uac70\ub4ed\uc81c\uacf1\uae09\uc218 \ud615\ud0dc\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\sin x = x &#8211; \\frac{x^3}{3!} + \\frac{x^5}{5!} &#8211; \\frac{x^7}{7!} + &#8211; \\cdots\\]<br \/>\n\uc774\ubbc0\ub85c, \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\sin x^2 = x^2 -\\frac{x^6}{3!} + \\frac{x^{10}}{5!} &#8211; \\frac{x^{14}}{7!} + &#8211; \\tag{2.2}\\cdots\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c \uc591\ubcc0\uc758 \ubd80\uc815\uc801\ubd84\uc744 \uad6c\ud558\uba74<br \/>\n\\[\\int \\sin x^2 \\,dx = C + \\frac{x^3}{3} &#8211; \\frac{x^7}{7 \\cdot 3!} + \\frac{x^{11}}{11\\cdot 5!} &#8211; \\frac{x^{15}}{15\\cdot 7!} + &#8211; \\cdots\\tag{2.3}\\]<br \/>\n\uc774\ub2e4. \ubd80\uc815\uc801\ubd84\uc744 \uac70\ub4ed\uc81c\uacf1\uae09\uc218 \ud615\ud0dc\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2eb\ud78c \ud615\ud0dc\uac00 \uc544\ub2c8\uae30 \ub54c\ubb38\uc5d0 \ud06c\uac8c \uc4f8\ubaa8\uac00 \uc5c6\uc744 \uac83 \uac19\uc9c0\ub9cc \uadf8\ub807\uc9c0 \uc54a\ub2e4. \uc5ec\ub7ec \uc751\uc6a9 \ubd84\uc57c\uc5d0\uc11c\ub294 \uc801\ubd84\uc758 \uadfc\uc0bf\uac12\uc744 \uacc4\uc0b0\ud558\ub294 \uac83\uc73c\ub85c\ub3c4 \ucda9\ubd84\ud560 \ub54c\uac00 \ub9ce\uae30 \ub54c\ubb38\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \ub2e4\uc74c \uc815\uc801\ubd84\uc744 \uacc4\uc0b0\ud574 \ubcf4\uc790.<br \/>\n\\[\\int_0^1 \\sin x^2 \\,dx\\tag{2.4}\\]<br \/>\n\ubd80\uc815\uc801\ubd84 (2.2)\uc5d0 \\(x=1,\\) \\(x=0\\)\uc744 \ub300\uc785\ud558\uc5ec \uacc4\uc0b0\ud55c \ud6c4 \ube7c\uba74<br \/>\n\\[\\int \\sin x^2\\,dx = \\frac{1}{3} &#8211; \\frac{1}{7\\cdot 3!} + \\frac{1}{11\\cdot 5!} &#8211; \\frac{1}{15\\cdot 7!} + &#8211; \\cdots\\tag{2.5}\\]<br \/>\n\uc774\ub2e4. \uc774 \uc815\uc801\ubd84\uc758 \uac12\uc744 \uc624\ucc28\uac00 \\(0.001\\) \uc774\ud558\uac00 \ub418\ub3c4\ub85d \uad6c\ud558\uace0 \uc2f6\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uad50\ub300\uae09\uc218\uc758 \uc624\ucc28 \ucd94\uc815 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\frac{1}{7 \\cdot 3!} &#038;= \\frac{1}{42} > 0.001 ,\\\\[4pt]<br \/>\n\\frac{1}{11\\cdot 5!} &#038;=\\frac{1}{1320} < 0.001\n\\end{align}\\]\n\uc774\ubbc0\ub85c\n\\[\\int_0^1 \\sin x^2 \\,dx \\approx \\frac{1}{3} - \\frac{1}{7 \\cdot 3!} \\approx 0.310\\]\n\uc744 \uc5bb\ub294\ub2e4. \ub9cc\uc57d \uc624\ucc28\uac00 \\(0.00001\\) \uc774\ud558\uac00 \ub418\ub3c4\ub85d \uad6c\ud558\uace0 \uc2f6\ub2e4\uba74 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c\n\\[\\int_0^1 \\sin x^2\\,dx \\approx \\frac{1}{3} - \\frac{1}{7\\cdot 3!} + \\frac{1}{11\\cdot 5!}  - \\frac{1}{15\\cdot 7!} \\approx 0.310268303\\]\n\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"leibnizsformula\"><\/a><\/p>\n<h3>\ub77c\uc774\ud504\ub2c8\uce20 \uacf5\uc2dd<\/h3>\n<p>\uc6d0\uc8fc\uc728\uc744 \uacc4\uc0b0\ud558\ub294 \ub2e4\uc74c \uacf5\uc2dd\uc744 <span class=\"defined\">\ub77c\uc774\ud504\ub2c8\uce20 \uacf5\uc2dd<\/span>(Leibniz&#8217;s formula)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<br \/>\n\\[\\frac{\\pi}{4} = 1 &#8211; \\frac{1}{3} + \\frac{1}{5} &#8211; \\frac{1}{7} + &#8211; \\cdots \\tag{3.1}\\]<br \/>\n\ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uc774 \uacf5\uc2dd\uc744 \uc774\ub04c\uc5b4\ub0b4\ubcf4\uc790. \\(-1 \\le x \\le 1\\)\uc77c \ub54c<br \/>\n\\[\\frac{d}{dx} \\arctan x = \\frac{1}{1+x^2}\\tag{3.2}\\]<br \/>\n\uc774\ub2e4. \\( -1 < x < 1\\)\uc77c \ub54c, \uc774 \uc2dd\uc758 \uc6b0\ubcc0\uc744 \ubb34\ud55c\ub4f1\ube44\uae09\uc218\ub85c \ub098\ud0c0\ub0b4\uba74\n\\[\\frac{d}{dx} \\arctan x = 1 - x^2 + x^4 - x^6 + - \\cdots \\tag{3.3}\\]\n\uc774\ub2e4. \ub2e4\uc2dc \uc774 \uc2dd\uc758 \uc591\ubcc0\uc758 \ubd80\uc815\uc801\ubd84\uc744 \uad6c\ud558\uba74\n\\[\\arctan x = x - \\frac{x^3}{3} + \\frac{x^5}{5} - \\frac{x^7}{7} + - \\cdots \\tag{3.4}\\]\n\uc774\ub2e4. \\(\\lvert x \\rvert < 1\\)\uc77c \ub54c (3.3)\uc758 \uc6b0\ubcc0\uc774 \uc218\ub834\ud558\uc9c0\ub9cc \\(x=1\\)\uc77c \ub54c\uc5d0\ub294 (3.3)\uc758 \uc6b0\ubcc0\uc774 \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c, \uc9c0\uae08\uae4c\uc9c0 \ub17c\uc758\ud55c \uacb0\uacfc\ub9cc\uc73c\ub85c\ub294 \\(x=1\\)\uc77c \ub54c (3.4)\uc758 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \ubcf4\uc7a5\ud560 \uc218 \uc5c6\ub2e4. (\ube44\ub85d (3.4)\uc758 \uc6b0\ubcc0\uc774 \uc218\ub834\ud568\uc5d0\ub3c4 \ubd88\uad6c\ud558\uace0 \ub9d0\uc774\ub2e4.) \uadf8\ub7ec\ub098 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Abel%27s_theorem\">\uc544\ubca8\uc758 \uc815\ub9ac<\/a>\ub97c \uc774\uc6a9\ud558\uba74 \\(x=1\\)\uc77c \ub54c (3.4)\uc758 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.<\/p>\n<p>(3.4)\uc758 \uc6b0\ubcc0\uc744 \\(T(x)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 (3.4)\ub294 \\(x = \\pm 1\\)\uc77c \ub54c \uc218\ub834\ud558\ubbc0\ub85c(\uad50\ub300\uae09\uc218 \ud310\uc815\ubc95), \\(T(x)\\)\ub294 \\(-1 \\le x \\le 1\\)\uc778 \ubc94\uc704\uc5d0\uc11c \uc815\uc758\ub418\uace0, \uc774 \ubc94\uc704\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud568\uc218\uc774\ub2e4. \ub610\ud55c \\(\\arctan\\)\ub3c4 \\(-1 \\le x \\le 1\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\begin{gather}<br \/>\n\\arctan 1 = \\lim_{x\\to 1^-} \\arctan x = \\lim_{x\\to 1^-} T(x) = T(1),\\\\[7pt]<br \/>\n\\arctan (-1) = \\lim_{x\\to -1^+} \\arctan x = \\lim_{x\\to -1^+} T(x) = T(-1)<br \/>\n\\end{gather}\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc989 \\(x= \\pm 1\\)\uc77c \ub54c\uc5d0\ub3c4 (3.4)\uc758 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4. (3.4)\uc758 \uc591\ubcc0\uc5d0 \\(x=1\\)\uc744 \ub300\uc785\ud558\uba74<br \/>\n\\[\\frac{\\pi}{4} = \\arctan 1 = 1 &#8211; \\frac{1}{3} + \\frac{1}{5} &#8211; \\frac{1}{7} + &#8211; \\cdots\\]<br \/>\n\ub85c\uc11c \uc6b0\ub9ac\uac00 \uc6d0\ud558\ub294 \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<p>\uc544\ubca8\uc758 \uc815\ub9ac\ub97c \uc774\uc6a9\ud558\uc9c0 \uc54a\uace0\uc11c\ub3c4 \\(x = \\pm 1\\)\uc77c \ub54c (3.4)\uac00 \uc131\ub9bd\ud568\uc744 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4. \ubb34\ud55c\ub4f1\ube44\uae09\uc218\uc758 \ud569 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud558\uba74<br \/>\n\\[\\frac{1}{1+t^2} = 1 &#8211; t^2 + t^4 &#8211; t^6 + &#8211; \\cdots + (-1)^n t^{2n} + \\frac{(-1)^{2n+1} t^{2n+2}}{1+t^2}\\]<br \/>\n\uc774\ub2e4. \uc88c\ubcc0\uacfc \uc6b0\ubcc0\uc774 \ubaa8\ub450 \ub2eb\ud78c \ud615\ud0dc(\uc720\ub9ac\uc2dd)\uc774\ubbc0\ub85c \uc774 \ub4f1\uc2dd\uc740 \\(t = \\pm 1\\)\uc77c \ub54c\uc5d0\ub3c4 \uc131\ub9bd\ud55c\ub2e4. \uc774 \uc2dd\uc758 \uc591\ubcc0\uc758 \ubd80\uc815\uc801\ubd84\uc744 \uad6c\ud558\uba74<br \/>\n\\[\\arctan x = x &#8211; \\frac{x^3}{3} + \\frac{x^5}{5} &#8211; \\frac{x^7}{7} +- \\cdots +(-1)^n \\frac{x^{2n+1}}{2n+1} + \\int_0^x \\frac{(-1)^{n+1} t^{2n+2}}{1+t^2} dt \\tag{3.5}\\]<br \/>\n\uc774\ub2e4. (3.5)\uc758 \ub9c8\uc9c0\ub9c9 \ud56d\uc5d0 \uc788\ub294 \uc801\ubd84\uc744 \\(R_n (x)\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \ud53c\uc801\ubd84\ud568\uc218\uc758 \ubd84\ubaa8\uac00 \\(1\\) \uc774\uc0c1\uc774\ubbc0\ub85c<br \/>\n\\[\\lvert R_n (x) \\rvert \\le \\int_0 ^{\\lvert x \\rvert} t^{2n+2} dt = \\frac{\\lvert x \\rvert ^{2n+3}}{2n+3}\\tag{3.6}\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(\\lvert x \\rvert \\le 1\\)\uc774\uba74 \\(n\\to \\infty\\)\uc77c \ub54c (3.6)\uc758 \ub9c8\uc9c0\ub9c9 \uc2dd\uc740 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc774 \uacb0\uacfc\uc640 (3.5)\ub97c \uacb0\ud569\ud558\uba74 \\(\\lvert x \\rvert \\le 1\\)\uc77c \ub54c<br \/>\n\\[\\arctan x = x &#8211; \\frac{x^3}{3} + \\frac{x^5}{5} &#8211; \\frac{x^7}{7} + &#8211; \\cdots\\]<br \/>\n\uc774 \uc131\ub9bd\ud568\uc744 \uc54c \uc218 \uc788\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"indeterminatelimit\"><\/a><\/p>\n<h3>\ubd80\uc815\ud615 \uadf9\ud55c\uc758 \uacc4\uc0b0<\/h3>\n<p>\ubd80\uc815\ud615 \uadf9\ud55c\uc744 \uacc4\uc0b0\ud560 \ub54c \uc790\uc8fc \uc0ac\uc6a9\ud558\ub294 \ubc29\ubc95\uc73c\ub85c <a href=\"https:\/\/sasamath.com\/blog\/articles\/calculus-indeterminate-forms-and-lhopitals-rule\/\">\ub85c\ud53c\ud0c8\uc758 \ubc95\uce59<\/a>\uc774 \uc788\ub2e4. \uadf8\ub7ec\ub098 \uacbd\uc6b0\uc5d0 \ub530\ub77c\uc11c\ub294 \ub85c\ud53c\ud0c8\uc758 \ubc95\uce59\uc744 \uc774\uc6a9\ud558\uc9c0 \uc54a\uace0\uc11c\ub3c4 \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \ubd80\uc815\ud615 \uadf9\ud55c\uc744 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.<\/span><\/p>\n<p>\ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud574 \ubcf4\uc790.<br \/>\n\\[\\lim_{x\\to 1} \\frac{\\ln x}{x-1}.\\]<br \/>\n\uc790\uc5f0\ub85c\uadf8\ud568\uc218\uc758 \ud14c\uc77c\ub7ec\uae09\uc218\ub294 \\( -1 < x < 1\\)\uc77c \ub54c\n\\[\\ln (1+x) = x - \\frac{x^2}{2} + \\frac{x^3}{2^2} - \\frac{x^4}{2^3} + - \\cdots\\]\n\uc774\ubbc0\ub85c, \\(0 < x < 2\\)\uc77c \ub54c\n\\[\\ln x = (x-1) - \\frac{1}{2} (x-1)^2 + \\cdots \\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c\n\\[\\lim_{x\\to 1}\\frac{\\ln x}{x-1} = \\lim_{x\\to 1}\\left( 1- \\frac{1}{2} (x-1) + \\cdots \\right) = 1\\]\n\uc774\ub2e4.\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p>\uc704 \ubcf4\uae30 2\uc5d0\uc11c\ucc98\ub7fc \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \ud45c\ud604\ub41c \ud568\uc218\uc758 \uadf9\ud55c\uc744 \uad6c\ud560 \ub54c \\(x\\)\uc5d0 \uac12\uc744 \uc9c1\uc811 \ub300\uc785\ud560 \uc218 \uc788\ub294 \uc774\uc720\ub294 \uc544\ubca8\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \uc815\uc758\ub41c \ud568\uc218\uac00 \uc5f0\uc18d\ud568\uc218\uc774\uae30 \ub54c\ubb38\uc774\ub2e4. \uc989<br \/>\n\\[f(x) = \\sum_{n=0}^{\\infty} a_n (x-a)^n\\]<br \/>\n\uc758 \uc6b0\ubcc0\uc774 \uc218\ub834\ud558\ub3c4\ub85d \ud558\ub294 \\(x\\)\uc758 \uc9d1\ud569\uc774 \\(I\\)\uc774\uace0 \\(t\\in I\\)\uc774\uba74<br \/>\n\\[\\lim_{\\substack{x\\to t\\\\ x\\in I}} f(x) = f(t)\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\lim_{\\substack{x\\to t\\\\ x\\in I}}\\sum_{n=0}^{\\infty} a_n (x-a)^n = \\sum_{n=0}^{\\infty} \\lim_{\\substack{x\\to t\\\\ x\\in I}} a_n (x-a)^n\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3.<\/span><\/p>\n<p>\ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud574 \ubcf4\uc790.<br \/>\n\\[\\lim_{x\\to 0}\\frac{\\sin x &#8211; \\tan x}{x^3}.\\]<br \/>\n\uc0ac\uc778 \ud568\uc218\uc640 \ud0c4\uc820\ud2b8 \ud568\uc218\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uc774\uc6a9\ud558\uba74<br \/>\n\\[\\sin x &#8211; \\tan x = -\\frac{x^3}{2} &#8211; \\frac{x^5}{8} &#8211; \\cdots = x^3 \\left( -\\frac{1}{2} &#8211; \\frac{x^2}{8} &#8211; \\cdots \\right)\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\lim_{x\\to 0}\\frac{\\sin x &#8211; \\tan x}{x^3} = \\lim_{x\\to 0}\\left( &#8211; \\frac{1}{2} &#8211; \\frac{x^2}{8} &#8211; \\cdots \\right) = &#8211; \\frac{1}{2}\\]<br \/>\n\uc774\ub2e4.<br \/>\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.<\/span><\/p>\n<p>\ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud574 \ubcf4\uc790.<br \/>\n\\[\\lim_{x\\to 0}\\left( \\frac{1}{\\sin x} &#8211; \\frac{1}{x} \\right).\\]<br \/>\n\uc0ac\uc778 \ud568\uc218\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uc774\uc6a9\ud558\uba74<br \/>\n\\[\\begin{align}<br \/>\n\\frac{1}{\\sin x} &#8211; \\frac{1}{x}<br \/>\n&#038;= \\frac{x-\\sin x}{x\\sin x} \\\\[3pt]<br \/>\n&#038;= \\frac{x &#8211; \\left( x &#8211; \\frac{x^3}{3!} + \\frac{x^5}{5!} &#8211; \\cdots \\right)}{x \\left( x &#8211; \\frac{x^3}{3!} + \\frac{x^5}{5!} &#8211; \\cdots \\right)} \\\\[3pt]<br \/>\n&#038;= \\frac{x^3 \\left( \\frac{1}{3!} &#8211; \\frac{x^2}{5!} + \\cdots \\right)}{x^2 \\left( 1- \\frac{x^2}{3!} + \\cdots \\right)} \\\\[3pt]<br \/>\n&#038;= x\\cdot \\frac{\\frac{1}{3!} &#8211; \\frac{x^2}{5!} + \\cdots }{1 &#8211; \\frac{x^2}{3!} + \\cdots }<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\lim_{x\\to 0} \\left( \\frac{1}{\\sin x} &#8211; \\frac{1}{x} \\right) = \\lim_{x\\to 0} \\left( x\\cdot \\frac{\\frac{1}{3!} &#8211; \\frac{x^2}{5!} + \\cdots }{1 &#8211; \\frac{x^2}{3!} + \\cdots } \\right) = 0\\]<br \/>\n\uc774\ub2e4.<br \/>\n<span class=\"qee\"><\/span>\n<\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### --><br \/>\n<a name=\"eulersidentity\"><\/a><\/p>\n<h3>\ud568\uc218\uc758 \uc815\uc758\uc5ed \ud655\uc7a5\ud558\uae30<\/h3>\n<p>\uc790\uc5f0\uc9c0\uc218\ud568\uc218 \\(\\exp x,\\) \uc0ac\uc778 \ud568\uc218 \\(\\sin x,\\) \ucf54\uc0ac\uc778 \ud568\uc218 \\(\\cos x,\\)\ub294 \ubaa8\ub4e0 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uc790\uc2e0\uc758 \ud14c\uc77c\ub7ec \uae09\uc218\uc640 \uc77c\uce58\ud558\ubbc0\ub85c, \uc774\ub4e4 \ud568\uc218\ub294 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub85c \ub300\uccb4\ud560 \uc218 \uc788\ub2e4(<a href=\"https:\/\/sasamath.com\/blog\/articles\/calculus-analytic-definition-of-exponential-function\/\">\ucc38\uace0 1<\/a>, <a href=\"https:\/\/sasamath.com\/blog\/articles\/calculus-analytic-definition-of-trigonometric-functions\/\">\ucc38\uace0 2<\/a>). \uc774\uc640 \uac19\uc740 \ud2b9\uc9d5\uc744 \uc774\uc6a9\ud558\uc5ec \uc774 \ud568\uc218\ub4e4\uc758 \uc815\uc758\uc5ed\uc744 \ubcf5\uc18c\uc218 \ubc94\uc704\ub85c \ud655\uc7a5\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\\(i\\)\uac00 \ud5c8\uc218 \ub2e8\uc704\ub97c \ub098\ud0c0\ub0b8\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uac00 \ubcf5\uc18c\uc218 \ubc94\uc704\uc5d0\uc11c <span class=\"defined\">\uc815\uc758\ub418\uc5b4 \uc788\ub2e4\uace0 \uac04\uc8fc<\/span>\ud558\uc790. \uadf8\ub7ec\uba74 \\(\\theta\\)\uac00 \uc2e4\uc218\uc77c \ub54c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\ne^{i\\theta}<br \/>\n&#038;= 1 + \\frac{i\\theta}{1!} + \\frac{(i\\theta )^2}{2!} + \\frac{(i\\theta )^3}{3!} + \\frac{(i\\theta )^4}{4!} + \\frac{(i\\theta )^5}{5!} + \\frac{(i\\theta )^6}{6!} + \\cdots \\\\[4pt]<br \/>\n&#038;= \\left( 1 &#8211; \\frac{\\theta ^2}{2!} + \\frac{\\theta^4}{4!} &#8211; \\frac{\\theta^6}{6!} + &#8211; \\cdots \\right) + i \\left( \\theta &#8211; \\frac{\\theta ^3}{3!} + \\frac{\\theta^5}{5!} &#8211; \\frac{\\theta ^7}{7!} + &#8211; \\cdots \\right) \\\\[6pt]<br \/>\n&#038;= \\cos \\theta + i \\sin \\theta<br \/>\n\\end{align}\\]<br \/>\n\uc774\uac83\uc740 \ub4f1\uc2dd \\(e^{i\\theta} = \\cos \\theta + i \\sin \\theta\\)\ub97c \uc99d\uba85\ud55c \uac83\uc774 <span class=\"defined\">\uc544\ub2c8\ub2e4<\/span>. \uc65c\ub0d0\ud558\uba74 \ud5c8\uc218 \uc9c0\uc218\ub97c \uc544\uc9c1 \uc815\uc758\ud558\uc9c0 \uc54a\uc558\uae30 \ub54c\ubb38\uc774\ub2e4. \ub300\uc2e0 \uc774 \ub4f1\uc2dd\uc744 \ud5c8\uc218\uc9c0\uc218\uc758 \uc815\uc758\ub85c \uc0bc\uae30\ub85c \ud55c\ub2e4. \uc989 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 3. (\ubcf5\uc18c\uc218 \uc9d1\ud569\uc744 \uc815\uc758\uc5ed\uc73c\ub85c \ud558\ub294 \uc9c0\uc218\ud568\uc218)<\/span><\/p>\n<p>\\(i\\)\uac00 \ud5c8\uc218 \ub2e8\uc704\uc774\uace0 \\(\\theta\\)\uac00 \uc2e4\uc218\uc77c \ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[e^{i\\theta} = \\cos \\theta + i\\sin\\theta .\\tag{5.1}\\]<br \/>\n\ub610\ud55c \\(a\\)\uac00 \uc591\uc218\uc774\uace0 \\(x\\)\uc640 \\(y\\)\uac00 \uc2e4\uc218\uc77c \ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[a^{x + iy} = a^x e^{iy \\ln a} .\\tag{5.2}\\]\n<\/p>\n<\/div>\n<p>\ub4f1\uc2dd (5.1)\uc744 <span class=\"defined\">\uc624\uc77c\ub7ec\uc758 \ud56d\ub4f1\uc2dd<\/span>(Euler&#8217;s identity)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub4f1\uc2dd (5.2)\ub294 \uc9c0\uc218\uac00 \ubcf5\uc18c\uc218\uc77c \ub54c \uc9c0\uc218 \ubc95\uce59\uc774 \uc131\ub9bd\ud55c\ub2e4\ub294 \uac00\uc815 \ud558\uc5d0 (5.1)\uc744 \uc790\uc5f0\uc2a4\ub7fd\uac8c \ud655\uc7a5\ud55c \uac83\uc774\ub2e4. \uc774\uc640 \uac19\uc774 \uae30\uc874\uc758 \ubc95\uce59\uc774 \ubcf4\uc874\ub418\ub3c4\ub85d \ud568\uc218\uc758 \uc815\uc758\uc5ed\uc744 \ud655\uc7a5\ud558\ub294 \ubc29\ubc95\uc744 <span class=\"defined\">\ud615\uc2dd\ubd88\uc5ed\uc758 \uc6d0\ub9ac<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>(5.1)\uc740 \uc9c0\uc218\ud568\uc218\uc758 \uc815\uc758\uc5ed\uc744 \ubcf5\uc18c\uc218 \uc9d1\ud569\uc73c\ub85c \ud655\uc7a5\ud55c \uac83\uc774\ub2e4. (5.1)\uc744 \uc774\uc6a9\ud558\uba74 \uc0ac\uc778\uacfc \ucf54\uc0ac\uc778\uc758 \uc815\uc758\uc5ed\uc744 \ubcf5\uc18c\uc218 \uc9d1\ud569\uc73c\ub85c \ud655\uc7a5\ud560 \uc218 \uc788\ub2e4. \uc989 \\(\\theta\\)\uac00 \uc2e4\uc218\uc77c \ub54c<br \/>\n\\[\\begin{align}<br \/>\ne^{i \\theta} &#038;= \\cos \\theta + i \\sin\\theta, \\tag{5.3}\\\\[7pt]<br \/>\ne^{-i \\theta} &#038;= \\cos \\theta &#8211; i\\sin\\theta \\tag{5.4}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. (5.3)\uacfc (5.4)\uc744 \ubcc0\ub9c8\ub2e4 \ub354\ud558\uc5ec \ub9cc\ub4e4\uc5b4\uc9c4 \ub4f1\uc2dd\uc758 \uc591\ubcc0\uc744 \\(2\\)\ub85c \ub098\ub204\uba74<br \/>\n\\[\\cos \\theta = \\frac{e^{i\\theta} + e^{-i\\theta}}{2}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4. \ub610\ud55c (5.3)\uc5d0\uc11c (5.4)\ub97c \ubcc0\ub9c8\ub2e4 \ube80 \ub4a4 \ub9cc\ub4e4\uc5b4\uc9c4 \ub4f1\uc2dd\uc758 \uc591\ubcc0\uc744 \\(2i\\)\ub85c \ub098\ub204\uba74<br \/>\n\\[\\sin \\theta = \\frac{e^{i\\theta} &#8211; e^{-i\\theta}}{2i}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc815\uc758\uc5ed\uc774 \ubcf5\uc18c\uc218 \uc9d1\ud569\uc778 \uc0bc\uac01\ud568\uc218\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 4. (\ubcf5\uc18c\uc218 \uc9d1\ud569\uc744 \uc815\uc758\uc5ed\uc73c\ub85c \ud558\ub294 \uc0bc\uac01\ud568\uc218)<\/span><\/p>\n<p>\\(i\\)\uac00 \ud5c8\uc218 \ub2e8\uc704\uc774\uace0 \\(z\\)\uac00 \ubcf5\uc18c\uc218\uc77c \ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\cos z &#038;= \\frac{e^{iz} + e^{-iz}}{2}, \\tag{5.5} \\\\[7pt]<br \/>\n\\sin z &#038;= \\frac{e^{iz} &#8211; e^{-iz}}{2i}. \\tag{5.6}<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<p>\\(x\\)\uac00 \uc2e4\uc218\uc77c \ub54c, \\(f(x) = e^{ix}\\)\ub85c \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\ub294 \uc8fc\uae30\uac00 \\(2\\pi\\)\uc778 \uc8fc\uae30\ud568\uc218\uc774\ub2e4. \uc989 \uc815\uc758\uc5ed\uc774 \ubcf5\uc18c\uc218 \uc9d1\ud569\uc778 \uc9c0\uc218\ud568\uc218\ub294 \uc77c\ub300\uc77c \ud568\uc218\uac00 \uc544\ub2c8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc9c0\uc218\ud568\uc218\uc758 \uc5ed\ud568\uc218\ub97c \uc815\uc758\ud560 \uc218 \uc5c6\ub2e4. \uc989 \ub85c\uadf8\ud568\uc218\ub97c \uc815\uc758\ud560 \uc218 \uc5c6\ub2e4. \ub300\uc2e0 \uc9c0\uc218\ud568\uc218\uc758 \uc815\uc758\uc5ed\uc744<br \/>\n\\[D = \\left\\{ x + yi \\,\\vert\\, x\\in\\mathbb{R} ,\\, 0 \\le y < 2\\pi \\right\\}\\]\n\uc640 \uac19\uc774 \uc81c\ud55c\ud558\uc5ec \uc77c\ub300\uc77c \ud568\uc218\uac00 \ub418\ub3c4\ub85d \ud568\uc73c\ub85c\uc368 \uadf8 \uc5ed\ud568\uc218\ub97c \ub85c\uadf8\ud568\uc218\ub85c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \ub610\ub294 \ud568\uc218\uc758 \uac1c\ub150\uc744 \ud655\uc7a5\ud558\uc5ec \uc5ec\ub7ec \uac1c\uc758 \uac12\uc744 \uac16\ub294 \ud568\uc218\ub97c \uc778\uc815\ud558\ub294 \ubc29\ubc95\uc774 \uc788\ub294\ub370, \uadf8\ub7ec\ud55c \ud568\uc218\ub97c <span class=\"defined\">\ub2e4\uac00\ud568\uc218<\/span>(multi-valued function)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uc774\uc640 \uad00\ub828\ub41c \uc790\uc138\ud55c \ub0b4\uc6a9\uc740 \ubcf5\uc18c\ud574\uc11d\ud559(complex analysis)\uc5d0\uc11c \ub2e4\ub8ec\ub2e4.<\/p>\n<p title=\"Source: https:\/\/www.acmescience.com\/2010\/02\/combinations-and-permutations-episode-36-master-of-us-all\/\"><img fetchpriority=\"high\" decoding=\"async\" alt=\"Euler equation\" longdesc=\"Euler equation is one of the most popular equation.\" src=\"\/blog\/wp-content\/uploads\/2019\/06\/euler_equation.gif\" alt=\"\" width=\"492\" height=\"329\" class=\"aligncenter size-full wp-image-4243\" \/><\/p>\n<p>\uc6d0\ub798 \ubc08\uc744 \ub123\uc73c\ub824\uace0 \ud588\uc73c\ub098 \uc624\uc77c\ub7ec \ub4f1\uc2dd\uc740 \uc6b0\uc544\ud558\uace0 \uc22d\uace0\ud558\uae30\uc5d0 \uacbd\uac74\ud558\uac8c \ud3ec\uc2a4\ud2b8\ub97c \ub9c8\uce5c\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### --><\/p>\n<h3><\/h3>\n<p><!--\n\n\n\n\n<div class=\"definition\">\n\n\n<p><span class=\"definition\">\uc815\uc758<\/span><\/p>\n\n\n\n\n<p> ... <\/p>\n\n<\/div>\n\n\n\n\n\n\n\n<div class=\"theorem\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac<\/span><\/p>\n\n\n\n\n<p> ... <\/p>\n\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85<\/p>\n\n\n\n\n<p>\n...\n<span class=\"qed\"><\/span><\/p>\n\n\n<\/div>\n\n\n\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 n.<\/span>\n...\n<span class=\"qee\"><\/span>\n<\/p>\n\n\n<\/div>\n\n\n\n\n\n\n<div class=\"remark\">\n\n\n<p><span class=\"remark\">\ucc38\uace0.<\/span>\n...\n<span class=\"qee\"><\/span>\n<\/p>\n\n\n<\/div>\n\n\n\n\n--><\/p>\n<p><!-- . --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc9c0\ub09c \ud3ec\uc2a4\ud2b8\uc5d0\uc11c \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \uc815\uc758\ud558\uace0 \ud568\uc218\ub97c \ud14c\uc77c\ub7ec \uae09\uc218\ub85c \ub098\ud0c0\ub0b4\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \ub610\ud55c \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uc0dd\uc131\ub41c \ud14c\uc77c\ub7ec \uae09\uc218\uac00 \\(f\\)\uc5d0 \uc218\ub834\ud568\uc744 \uc99d\uba85\ud558\ub294 \ubc29\ubc95\ub3c4 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \ub354\ubd88\uc5b4 \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uc0bc\uac01\ud568\uc218\ub97c \uc815\uc758\ud558\ub294 \ubc29\ubc95(\uad00\ub828 \ud3ec\uc2a4\ud2b8)\uacfc \uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uc9c0\uc218\ud568\uc218\ub97c \uc815\uc758\ud558\ub294 \ubc29\ubc95(\uad00\ub828 \ud3ec\uc2a4\ud2b8)\ub3c4 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ud14c\uc77c\ub7ec \uae09\uc218\ub97c \ud65c\uc6a9\ud55c \ub2e4\uc591\ud55c \uc608\ub97c \uc0b4\ud3b4\ubcf8\ub2e4. \ub0b4\uc6a9 \uc21c\uc11c \uc774\ud56d\uae09\uc218 \ube44\ucd08\ub4f1\uc801\ubd84 \ub77c\uc774\ud504\ub2c8\uce20 \uacf5\uc2dd \ubd80\uc815\ud615 \uadf9\ud55c\uc758 \uacc4\uc0b0 \ud568\uc218\uc758 \uc815\uc758\uc5ed \ud655\uc7a5\ud558\uae30 \ubbf8\ub9ac \uc54c\uc544\uc57c \ud560 \ub0b4\uc6a9 \uac70\ub4ed\uc81c\uacf1\uae09\uc218 (\uad00\ub828 \uae00) \ud14c\uc77c\ub7ec&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[370,365,224,366,364,150,367,368,362,361,363,223,360,369],"class_list":["post-2026","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-370","tag-365","tag-224","tag-366","tag-364","tag-150","tag-367","tag-368","tag-362","tag-361","tag-363","tag-223","tag-360","tag-369"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2026","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=2026"}],"version-history":[{"count":111,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2026\/revisions"}],"predecessor-version":[{"id":5966,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2026\/revisions\/5966"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2026"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2026"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2026"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}