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Residue
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Residue

Residue[expr,{z,z0}]

求 expr 在点 z=z0 处的留数.

更多信息和选项

  • 留数定义为在 expr 的罗朗展开中的 (z-z0)^-1 的系数.
  • Wolfram 语言通常仅在能计算幂级数的点上求得留数.

范例

打开所有单元关闭所有单元

基本范例  (2)常见实例总结

In[1]:=1
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https://wolfram.com/xid/0c129jq-qr2
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0c129jq-v0u
Out[2]=2
In[1]:=1
✖
https://wolfram.com/xid/0c129jq-6a
Out[1]=1

范围  (3)标准用法实例范围调查

In[1]:=1
✖
https://wolfram.com/xid/0c129jq-eje
Out[1]=1
In[1]:=1
✖
https://wolfram.com/xid/0c129jq-rbb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0c129jq-sbm
Out[2]=2
In[1]:=1
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https://wolfram.com/xid/0c129jq-psu
Out[1]=1

应用  (1)用该函数可以解决的问题范例

对于复数函数的积分举例说明柯西定理：

In[1]:=1
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https://wolfram.com/xid/0c129jq-hrv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0c129jq-db3
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0c129jq-gz
Out[3]=3

属性和关系  (1)函数的属性及与其他函数的关联

用 FunctionPoles 求函数的极点：

In[1]:=1
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https://wolfram.com/xid/0c129jq-hvf9j1
Out[1]=1

用 Residue 求极点处的留数：

In[2]:=2
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https://wolfram.com/xid/0c129jq-pvc44t
Out[2]=2

ResidueSum 给出所有极点处留数的和：

In[3]:=3
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https://wolfram.com/xid/0c129jq-7bwxf0
Out[3]=3

可能存在的问题  (1)常见隐患和异常行为

在分支点处没有留数：

In[1]:=1
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https://wolfram.com/xid/0c129jq-gbd
Out[1]=1
Wolfram Research (1991)，Residue，Wolfram 语言函数，https://reference.wolfram.com/language/ref/Residue.html.
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Wolfram Research (1991)，Residue，Wolfram 语言函数，https://reference.wolfram.com/language/ref/Residue.html.

文本

Wolfram Research (1991)，Residue，Wolfram 语言函数，https://reference.wolfram.com/language/ref/Residue.html.

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Wolfram Research (1991)，Residue，Wolfram 语言函数，https://reference.wolfram.com/language/ref/Residue.html.

CMS

Wolfram 语言. 1991. "Residue." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/Residue.html.

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Wolfram 语言. 1991. "Residue." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/Residue.html.

APA

Wolfram 语言. (1991). Residue. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/Residue.html 年

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Wolfram 语言. (1991). Residue. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/Residue.html 年

BibTeX

@misc{reference.wolfram_2025_residue, author="Wolfram Research", title="{Residue}", year="1991", howpublished="\url{https://reference.wolfram.com/language/ref/Residue.html}", note=[Accessed: 04-April-2025 ]}

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@misc{reference.wolfram_2025_residue, author="Wolfram Research", title="{Residue}", year="1991", howpublished="\url{https://reference.wolfram.com/language/ref/Residue.html}", note=[Accessed: 04-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_residue, organization={Wolfram Research}, title={Residue}, year={1991}, url={https://reference.wolfram.com/language/ref/Residue.html}, note=[Accessed: 04-April-2025 ]}

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@online{reference.wolfram_2025_residue, organization={Wolfram Research}, title={Residue}, year={1991}, url={https://reference.wolfram.com/language/ref/Residue.html}, note=[Accessed: 04-April-2025 ]}