证明复数的二项式定理 有关复数的证明
\uff08\u6570\u7269\u9898\u76ee\uff09\u5982\u4f55\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e\u590d\u6570\u7684\u4e8c\u9879\u5f0f\u5b9a\u7406\u6570\u5b66\u5f52\u7eb3\u6cd5\u4e0d\u662f\u683c\u5f0f\u5316\uff1f\u6309\u7167\u683c\u5f0f\u6765\u5c31\u884c\u4e86
\u8fd9\u4e2a\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u3002
\u5f53n=1\u65f6\uff0c\u5de6\u8fb9=cosx+isinx;
\u53f3\u8fb9=cosx+isinx
\u6545 n=1\u65f6\uff0c\u7ed3\u8bba\u6210\u7acb\u3002
\u5f53n=k\u65f6\uff0c\u8bbe\u7ed3\u8bba\u6210\u7acb\u3002\u5373\u6709\uff08cosx+isinx\uff09^k=cos(kx)+isin(kx)
\u5219\u5f53m=k+1\u65f6\uff0c\u53f3\u8fb9=cos((k+1)x)+isin((k+1)x);
\u5de6\u8fb9=(cosx+isinx)^(k+1)=((cosx+isinx\uff09^k)*(cosx+isinx)
=(cos(kx)+isin(kx))*(cosx+isinx)
=cos(kx)cosx-sin(kx)sinx+cos(kx)*isinx+isin(kx)*cosx
=cos((k+1)x)+isin((k+1)x)(\u7528\u5230\u4e86\u4e09\u89d2\u51fd\u6570\u548c\u7684\u5c55\u5f00\u5f0f)
这个用数学归纳法。当n=1时，左边=cosx+isinx;右边=cosx+isinx故n=1时，结论成立。当n=k时，设结论成立。即有（cosx+isinx）^k=cos(kx)+isin(kx)则当m=k+1时，右边=cos((k+1)x)+isin((k+1)x);左边=(cosx+isinx)^(k+1)=((cosx+isinx）^k)*(cosx+isinx)=(cos(kx)+isin(kx))*(cosx+isinx)=cos(kx)cosx-sin(kx)sinx+cos(kx)*isinx+isin(kx)*cosx=cos((k+1)x)+isin((k+1)x)(用到了三角函数和的式)
绛旓細褰搉=1鏃讹紝宸﹁竟=cosx+isinx;鍙宠竟=cosx+isinx鏁卬=1鏃讹紝缁撹鎴愮珛銆傚綋n=k鏃讹紝璁剧粨璁烘垚绔銆傚嵆鏈夛紙cosx+isinx锛塣k=cos(kx)+isin(kx)鍒欏綋m=k+1鏃讹紝鍙宠竟=cos((k+1)x)+isin((k+1)x);宸﹁竟=(cosx+isinx)^(k+1)=((cosx+isinx锛塣k)*(cosx+isinx)=(cos(kx)+isin(kx))*(cosx+isinx)=...
绛旓細鍖栫畝寰楋細n^2-3n-54=0銆傚氨鏄細锛坣-9锛*锛坣+6锛=0銆俷灏辨槸9鎴-6銆-6涓嶅悎棰樻剰鑸嶅幓銆傜嚎鎬у舰寮 濡傛灉浜岄」寮忕殑褰㈠紡涓篴x+b锛堝叾涓璦涓巄鏄父鏁帮紝x鏄彉閲忥級锛岄偅涔堣繖涓簩椤瑰紡鏄嚎鎬х殑銆傚鏁版槸褰㈠紡涓篴+bi鐨勪簩椤瑰紡锛屽叾涓璱鏄-1鐨勫钩鏂规牴銆備簩椤瑰紡瀹氱悊涓庢潹杈変笁瑙掑舰鏄竴瀵瑰ぉ鐒剁殑鏁板舰瓒ｉ亣锛屽畠鎶婃暟褰㈢粨鍚堝甫杩...
绛旓細1銆鍩烘湰鍏紡锛氫簩椤瑰紡瀹氱悊鐨勫熀鏈叕寮忓彲浠ョ敤濡備笅琛ㄧず锛锛坅+b锛塣n=C锛坣锛0锛*a^n*b^0+C锛坣锛1锛*a^锛坣-1锛*b^1+C锛坣锛2锛*a^锛坣-2锛*b^2...+C锛坣锛宯-1锛*a^1*b^锛坣-1锛+C锛坣锛宯锛*a^0*b^n銆2銆佸叾涓紝a銆乥鏄疄鏁版垨澶嶆暟锛宯鏄潪璐熸暣鏁般侰锛坣锛宬锛夎〃绀轰粠n涓厓绱...
绛旓細澶嶆暟鏄舰寮忎负a+bi鐨勪簩椤瑰紡锛屽叾涓璱鏄-1鐨勫钩鏂规牴銆
绛旓細浜岄」寮忓畾鐞嗗睍寮寮忓叕寮忎负锛^n = C脳a^n脳b^0 + C脳a^脳b^1 + C脳a^脳b^2 + 鈥 + C脳a^脳b^i + 鈥 + C脳a^0脳b^n銆傝缁嗚В閲婂涓嬶細浜岄」寮忓畾鐞嗘槸鏁板涓潪甯搁噸瑕佺殑涓涓叕寮锛岀敤浜庡睍寮褰㈠^n鐨勫紡瀛愩傝繖閲岀殑a鍜宐鏄换鎰忓疄鏁版垨澶嶆暟锛宯鏄潪璐熸暣鏁般傝瀹氱悊鐨勫睍寮寮忚〃绀轰簡灏嗕竴涓簩椤瑰紡...
绛旓細褰搉=1鏃讹紝宸﹁竟=cosx+isinx;鍙宠竟=cosx+isinx 鏁 n=1鏃讹紝缁撹鎴愮珛銆傚綋n=k鏃讹紝璁剧粨璁烘垚绔嬨傚嵆鏈夛紙cosx+isinx锛塣k=cos(kx)+isin(kx)鍒欏綋m=k+1鏃讹紝鍙宠竟=cos((k+1)x)+isin((k+1)x);宸﹁竟=(cosx+isinx)^(k+1)=((cosx+isinx锛塣k)*(cosx+isinx)=(cos(kx)+isin(kx))*(cosx+isinx...
绛旓細浜岄」寮忓畾鐞嗕腑鐨勬湁鐞嗛」鎰忔濓細绯绘暟涓烘湁鐞嗘暟锛屾鏁颁负鏁存暟鐨勯」鍙仛鏈夌悊椤广傛暣鏁板拰鍒嗘暟缁熺О涓烘湁鐞嗘暟銆備换浣曚竴涓湁鐞嗘暟閮藉彲浠ュ湪鏁拌酱涓婄殑鐐规潵琛ㄧず銆傛棤闄愪笉寰幆灏忔暟绉颁箣涓烘棤鐞嗘暟銆
绛旓細鐗涢】浠浜岄」寮忓畾鐞浣滀负鍩虹煶鍙戞槑鍑轰簡寰Н鍒嗐俒4] 鍏跺湪鍒濈瓑鏁板涓簲鐢ㄤ富瑕佸湪浜庝竴浜涚矖鐣ョ殑鍒嗘瀽鍜屼及璁′互鍙璇佹槑鎭掔瓑寮忕瓑銆傝繖涓畾鐞嗗湪閬椾紶瀛︿腑涔熸湁鍏剁敤姝︿箣鍦帮紝鍏蜂綋搴旂敤鑼冨洿涓猴細鎺ㄦ祴鑷氦鍚庝唬缇や綋鐨勫熀鍥犲瀷鍜屾鐜囥佹帹娴嬭嚜浜ゅ悗浠ｇ兢浣撶殑琛ㄧ幇鍨嬪拰姒傜巼銆佹帹娴嬫潅浜ゅ悗浠ｇ兢浣撶殑琛ㄧ幇鍨嬪垎甯冨拰姒傜巼銆侀氳繃娴嬩氦鍒嗘瀽鏉傚悎浣撹嚜浜ゅ悗浠ｇ殑...
绛旓細浜岄」寮忓畾鐞鎸囩殑鏄細浜岄」寮忓畾鐞嗭紝鍙堢О鐗涢】浜岄」寮忓畾鐞嗭紝鐢辫壘钀ㄥ厠路鐗涢】浜1664骞淬1665骞撮棿鎻愬嚭銆傝瀹氱悊缁欏嚭涓や釜鏁颁箣鍜岀殑鏁存暟娆″箓璇稿灞曞紑涓虹被浼奸」涔嬪拰鐨勬亽绛夊紡銆備簩椤瑰紡瀹氱悊鍙互鎺ㄥ箍鍒颁换鎰忓疄鏁版骞傦紝鍗冲箍涔変簩椤瑰紡瀹氥備簩椤瑰紡瀹氱悊鐨勬剰涔夛細鐗涢】浠ヤ簩椤瑰紡瀹氱悊浣滀负鍩虹煶鍙戞槑鍑轰簡寰Н鍒嗭紝鍏跺湪鍒濈瓑鏁板涓簲鐢ㄤ富瑕佸湪浜庝竴浜涚矖鐣...
绛旓細浜岄」寮忓畾鐞锛屽張绉扮墰椤夸簩椤瑰紡瀹氱悊锛岀敱鑹捐惃鍏嬄风墰椤夸簬1664-1665骞存彁鍑恒傚叕寮忎负:(a+b)^n=C(n,0)a^n+C(n,1)a^(n-1)b+...+C(n,i)a^(n-i)b^i+...+C(n,n)b^n 寮忎腑,C(n,i)琛ㄧず浠巒涓厓绱犱腑浠诲彇i涓殑缁勫悎鏁=n!/(n-i)!i!姝ゅ畾鐞嗘寚鍑:1銆(a+b)^n鐨勪簩椤灞曞紑寮忓叡鏈塶+1...
