電大復(fù)變函數(shù)與積分變換重點公式歸納

復(fù)變函數(shù)與積分變換復(fù)習(xí)提綱復(fù)變函數(shù)復(fù)變數(shù)和復(fù)變函數(shù)SKIPIF1<0復(fù)變函數(shù)的極限與連續(xù)極限SKIPIF1<0連續(xù)SKIPIF1<0解析函數(shù)復(fù)變函數(shù)SKIPIF1<0可導(dǎo)與解析的概念?？挛鳌杪匠陶莆绽肅-R方程SKIPIF1<0判別復(fù)變函數(shù)的可導(dǎo)性與解析性。掌握復(fù)變函數(shù)的導(dǎo)數(shù)：SKIPIF1<0初等函數(shù)重點掌握初等函數(shù)的計算和復(fù)數(shù)方程的求解。1、冪函數(shù)與根式函數(shù)SKIPIF1<0單值函數(shù)SKIPIF1<0(k=0、1、2、…、n-1)n多值函數(shù)2、指數(shù)函數(shù)：SKIPIF1<0性質(zhì)：（1）單值.（2）復(fù)平面上處處解析，SKIPIF1<0（3）以SKIPIF1<0為周期3、對數(shù)函數(shù)SKIPIF1<0（k=0、±1、±2……）性質(zhì)：（1）多值函數(shù),（2）除原點及負(fù)實軸處外解析,（3）在單值解析分枝上:SKIPIF1<0。4、三角函數(shù)：SKIPIF1<0SKIPIF1<0性質(zhì)：（1）單值（2）復(fù)平面上處處解析（3）周期性（4）無界5、反三角函數(shù)（了解）反正弦函數(shù)SKIPIF1<0反余弦函數(shù)SKIPIF1<0性質(zhì)與對數(shù)函數(shù)的性質(zhì)相同。6、一般冪函數(shù)：SKIPIF1<0（k=0、±1…）四、調(diào)和函數(shù)與共軛調(diào)和函數(shù)：1)調(diào)和函數(shù):SKIPIF1<02)已知解析函數(shù)的實部（虛部），求其虛部（實部）有三種方法：a）全微分法b）利用C-R方程c）不定積分法第三章解析函數(shù)的積分一、復(fù)變函數(shù)的積分SKIPIF1<0存在的條件。二、復(fù)變函數(shù)積分的計算方法1、沿路徑積分：SKIPIF1<0利用參數(shù)法積分，關(guān)鍵是寫出路徑的參數(shù)方程。2、閉路積分：a)SKIPIF1<0利用留數(shù)定理，柯西積分公式，高階導(dǎo)數(shù)公式。b)SKIPIF1<0利用參數(shù)積分方法三、柯西積分定理：SKIPIF1<0推論1：積分與路徑無關(guān)SKIPIF1<0推論2：利用原函數(shù)計算積分SKIPIF1<0推論3：二連通區(qū)域上的柯西定理SKIPIF1<0推論4：復(fù)連通區(qū)域上的柯西定理SKIPIF1<0四、柯西積分公式：SKIPIF1<0SKIPIF1<0五、高階導(dǎo)數(shù)公式：SKIPIF1<0SKIPIF1<0解析函數(shù)的兩個重要性質(zhì)：解析函數(shù)SKIPIF1<0在任一點SKIPIF1<0的值可以通過函數(shù)沿包圍點SKIPIF1<0的任一簡單閉合回路的積分表示。解析函數(shù)有任意階導(dǎo)數(shù)。本章重點：掌握復(fù)變函數(shù)積分的計算方法沿路徑積分SKIPIF1<01）利用參數(shù)法積分2）利用原函數(shù)計算積分。閉路積分SKIPIF1<0利用留數(shù)定理計算積分。第四章解析函數(shù)的級數(shù)一、冪級數(shù)及收斂半徑：SKIPIF1<01、一個收斂半徑為R（≠0）的冪級數(shù)，在收斂圓內(nèi)的和函數(shù)SKIPIF1<0是解析函數(shù)，在這個收斂圓內(nèi)，這個展開式可以逐項積分和逐項求導(dǎo)，即有：SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<02、收斂半徑的計算方法比值法：SKIPIF1<0根值法：SKIPIF1<0二、泰勒（Taylor）級數(shù)1、如函數(shù)SKIPIF1<0在圓域SKIPIF1<0內(nèi)解析，那么在此圓域內(nèi)SKIPIF1<0可以展開成Taylor級數(shù)SKIPIF1<0SKIPIF1<01）展開式是唯一的。故將函數(shù)在解析點的鄰域中展開冪級數(shù)一定是Taylor級數(shù)。2)收斂半徑是展開點到SKIPIF1<0的所有奇點的最短距離。3）展開式的系數(shù)可以微分計算：SKIPIF1<04）解析函數(shù)可以用Taylor級數(shù)表示。2、記住一些重要的泰勒級數(shù)：1）SKIPIF1<02）SKIPIF1<03）SKIPIF1<04）SKIPIF1<0三、羅蘭（Laurent）級數(shù)如果函數(shù)SKIPIF1<0在圓環(huán)城SKIPIF1<0內(nèi)解析，則SKIPIF1<0=SKIPIF1<0SKIPIF1<0（n=0、±1、±2……）1、展開式是唯一的，即只要把函數(shù)在圓環(huán)城內(nèi)展開為冪級數(shù)即為Laurent級數(shù)。2、展開式的系數(shù)是不可以利用積分計算。利用已知的冪級數(shù)，通過代數(shù)運算把函數(shù)展開成Laurent級數(shù)。3、注意展開的區(qū)域，在展開點的所有解析區(qū)域展開。四、孤立奇點1、定義：若b是SKIPIF1<0的孤立奇點，則SKIPIF1<0在SKIPIF1<0內(nèi)解析。在此點SKIPIF1<0可展開為羅蘭級數(shù)，SKIPIF1<0=SKIPIF1<02、分類：孤立奇點SKIPIF1<0把函數(shù)在奇點的去心鄰域中展開為羅蘭級數(shù)，求解C-13、極點留數(shù)計算a)如果b是SKIPIF1<0的一階極點，則SKIPIF1<0b)如果b是SKIPIF1<0的m階極點，則SKIPIF1<0c)如b是SKIPIF1<0的一階極點，且P(b)≠0，那么SKIPIF1<0d)SKIPIF1<0e)若SKIPIF1<0是SKIPIF1<0的可去奇點，并且SKIPIF1<0，SKIPIF1<0關(guān)系：全平面留數(shù)之和為零。SKIPIF1<0本章重點：函數(shù)展開成Taylor級數(shù)，并能寫出收斂半徑。函數(shù)在解析圓環(huán)城內(nèi)展開成Laurent級數(shù)。孤立奇點（包含SKIPIF1<0點）的判定及其留數(shù)的計算。第五章留數(shù)定理的應(yīng)用一、SKIPIF1<0條件：（1）R(sinq，cosq)為cosq與sinq的有理函數(shù)（2）R(?)在[0，2p]或者[-p，p]上連續(xù)。令SKIPIF1<0，則SKIPIF1<0，SKIPIF1<0，SKIPIF1<0。SKIPIF1<0SKIPIF1<0SKIPIF1<0注意留數(shù)是計算單位圓中的奇點。二、SKIPIF1<0條件：（1）SKIPIF1<0SKIPIF1<0是x的多項式。（2）SKIPIF1<0（3）分母階次比分子階次至少高二次則SKIPIF1<0SKIPIF1<0是SKIPIF1<0在上半平面的奇點。三、SKIPIF1<0（SKIPIF1<0）條件：（1）SKIPIF1<0，且SKIPIF1<0比SKIPIF1<0至少高一階，（2）SKIPIF1<0，（3）SKIPIF1<0SKIPIF1<0SKIPIF1<0SKIPIF1<0，SKIPIF1<0重點關(guān)注第一和第三種類型Fourier變換一、傅立葉變換SKIPIF1<0SKIPIF1<0SKIPIF1<0函數(shù)的傅立葉變換?SKIPIF1<0.SKIPIF1<0一些傅立葉變換及逆變換?SKIPIF1<0?SKIPIF1<0四、性質(zhì)：?SKIPIF1<0相似性質(zhì)?SKIPIF1<02、?SKIPIF1<0延遲性質(zhì)?SKIPIF1<0位移性質(zhì)3、微分性質(zhì)?SKIPIF1<0?SKIPIF1<0?SKIPIF1<0?SKIPIF1<04、積分性質(zhì)?SKIPIF1<0由Fourier變換的微分和積分性質(zhì)，我們可以利用Fourier變換求解微積分方程。四、卷積和卷積定理SKIPIF1<0?SKIPIF1<0?SKIPIF1<0＊五、三維Fourier變換及反演本章重點：利用定義計算Fourier變換第八章Laplace變換一、拉普拉斯變換?SKIPIF1<0二、幾個重要的拉普拉斯變換及逆變換?SKIPIF1<0?SKIPIF1<0?SKIPIF1<0?SKIPIF1<0?SKIPIF1<0?SKIPIF1<0?SKIPIF1<0?SKIPIF1<0?SKIPIF1<0?SKIPIF1<0四、拉普拉斯變換的性質(zhì)1、?SKIPIF1<02、?SKIPIF1<03、?SKIPIF1<0?SKIPIF1<0?SKIPIF1<04、?SKIPIF1<0?SKIPIF1<0五、卷積：SKIPIF1<0?SKIPIF1<0六、Laplace反演SKIPIF1<0七、Laplace逆變換（1）部分分式法（2）卷積定理（3）Laplace反演公式（留數(shù)定理）（4）利用Laplace變換的性質(zhì)八、利用Laplace變換求解微積分方程（1）對方程取Laplace變換，得到象函數(shù)的代數(shù)方程（2）解代數(shù)方程，得到像函數(shù)的表達(dá)式（3）求像函數(shù)的拉普拉斯逆變換微分方程像函數(shù)的代數(shù)方程像函數(shù)微分方程像函數(shù)的代數(shù)方程像函數(shù)像原函數(shù)解函數(shù)拉氏逆變換本章重點：利用定義和性質(zhì)計算Laplace變換。計算Laplace逆變換。利用Laplace變換求解微積分方程。aganemploymenttribunalclaiEmloymenttribunalssortoutdisagreementsbetweenemployersandemployees.Youmayneedtomakeaclaimtoanemploymenttribunalif:youdon'tagreewiththedisciplinaryactionyouremployerhastakenagainstyouyouremployerdismissesyouandyouthinkthatyouhavebeendismissedunfairly.Formoreinformu,takeadvicefromoneoftheorganisationslistedunder

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