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# §3 留数在定积分计算上的应用 - PowerPoint PPT Presentation

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§3 留数在定积分计算上的应用

1. 形如 的积分, 其中R(cosq,sinq )为 cosq与sinq 的有理函数.

1 计算 的值.

[解] 由于0<p<1, 被积函数的分母在0q  2p内不为零, 因

2 计算 的值.

3

CR

z3

z2

z1

O

x

-R

R

5

3. 形如 的积分

y

y

z3

CR

1

z2

y=sinq

z1

x

-R

O

R

p

O

q

6 计算 的值.

[解] 这里m=2,n=1,m-n=1.R(z)在实轴上无孤立奇点,因而所求的积分是存在的. 在上半平面内有一级极点ai,

4 计算积分 的值.

[解]因为 是偶函数, 所以

CR

Cr

-r

r

O

-R

R

x

j (z)在z=0处解析, 且j (0)=i, 当|z|充分小时可使|j (z)|2,