Understanding Poles and Zeros in Complex Analysis

In summary, the conversation discusses a question about the function 1/(z-w)^4 and its poles. There is confusion about whether the function has one pole of order 4 or four poles of order 1. It is clarified that the function has one pole at w. The conversation also touches on the terms singularity and pole, with the conclusion that a pole is one type of singularity.
  • #1
Ant farm
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Oh god, so confused and panicked today:cry:
I know this is a very basic question, but, givin the function 1/(z-w)^4

does this have one pole of order 4, or possibly 4 poles of order 1...?

Also, could you please clarify,
''to get the zero's of a function, set the numerator = 0''
''to get the poles of a function, set the denominator = 0''
is this correct??
 
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  • #2
(z-w)^4 is 0 iff z=w. So, it has one pole at w.

I always though when the denominator was 0 it was referred to as a singularity?
 
  • #3
as yes, right you are, and a pole is one of the three types of singularity... removable, pole and essential!
 

1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of complex numbers and their functions. It involves the use of techniques from calculus, algebra, and geometry to analyze and understand the properties of complex-valued functions.

2. What is the significance of the expression 1/(z-w)^4 in complex analysis?

In complex analysis, the expression 1/(z-w)^4 is known as a complex-valued function. It represents a function of two complex variables, z and w, where z is the independent variable and w is the dependent variable. This function is particularly useful in analyzing the behavior of complex functions near singular points.

3. How is the expression 1/(z-w)^4 used in solving complex analysis problems?

The expression 1/(z-w)^4 is often used in complex analysis to evaluate integrals, determine the analyticity of a function, and find the behavior of a function near singular points. It is also used in the development of Laurent series, which are used to express complex functions as infinite sums of powers of the complex variable z.

4. Can the expression 1/(z-w)^4 be simplified in complex analysis?

Yes, the expression 1/(z-w)^4 can be simplified in certain cases. For example, if w is a constant, the expression can be simplified to 1/z^4. Furthermore, if z and w are both complex numbers, the expression can be simplified using algebraic techniques.

5. What are some applications of the expression 1/(z-w)^4 in real-world problems?

The expression 1/(z-w)^4 has various applications in fields such as physics, engineering, and economics. In physics, it is used to solve problems related to fluid dynamics and electromagnetism. In engineering, it is used in the analysis of electric circuits and signal processing. In economics, it is used to model market behavior and analyze economic data.

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