武汉科技大学复变函数与积分变换试题A卷参考答案(2010年12月).doc

2010 年~ 2011年第一学期 复变函数与积分变换A卷参考答案
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一、单项选择题。（每小题3分，共15分）
1．A 2．Ｄ 3． C 4．Ｂ ５ B．
二、填空题。（每小题3分，共15分）
6． 7． 2 8． 0 9． 10．
三、计算题。（本题6分）
11、求和的值．
解： ························································································（2分）
·····························································································（3分）
···································································································（5分）
····································································································（6分）
四、计算下列复变函数的积分（本大题共3小题，每小题6分，共18分）
12．（1） （2）
（3）
解：（1）=··································（4分）
···········································································································（6分）（2）··········································································（4分）
····································································································（6分）（3）在曲线内，函数仅有一个奇点，且是它的一阶极点，由留数定理得：······························································································································（2分）
·········································································（4分）
····················································（6分）
五、解答题。（本大题共2小题，每小题6分，共12分）
13．试证函数是调和函数，并求函数，使得为解析函数，且满足．
解：由于，，显然，因此是调和函数．·（1分）
因为···························································（3分）
故········································································（5分）
又由得，，从而·················· （6分）4．利用留数计算积分．
解：由于函数在上半平面内只有一个极点，故 ···························································· （3分）
··················································································（5分）
·····························································································（6分）
六、(本大题14分)
16．将函数在处展开为泰勒级数．（6分）
解：··························································································（２分）
························································（4分）
·············································（6分）
17．将函数分别在圆环域（1）；（2）内展开为洛朗级数．（8分）
解：当时
··········································································（２分）
·····················································（４分）
当时
····································（6分）
····················································································（8分）
七、（本大题共2小题，每小题10分，共20分）
18．求函数的Laplace变换．
解：由于 L，······································································································· （2分）
根据延迟性质，得 L，······································································· （6分）
又根据相似性质，可得F ·········································（10分）

19．利用Laplace变换求解常微分方程．
解：原方程两边取Laplace变换，得，··································· （4分）
将带入得：
·································································································· （6分）

求拉氏逆变换得原方程的解为.·························（10分）

注：1、教师命题时题目之间不留空白； 2、考生不得在试题纸上答题，教师只批阅答题册正面部分，若考生须在试题图上作解答，请另附该试题图。3、请在试卷类型、考试方式后打勾注明。
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