How do you expand a function in Laurent series?
We will expand as a Laurent series on the annulus Ann ( 1 , 0 , ∞ ) .
We first change co-ordinates via w = z − 1 .
Then z = 1 + w and we are interested in expanding g ( w ) = exp ( 1 + w ) w 2 on Ann ( 0 , 0 , ∞ ) ..
Is the Laurent series expansion unique?
Show that an=bn for all n∈Z.
This means that the Laurent series expansion is unique.
Hint: It suffices to show that if f≡0, then an=0 for all n.
Use ∑∞n=0anzn=∑−1n=−∞−anzn to construct a bounded entire function..
What is Laurent's series expansion of complex function?
Laurent's series, also known as Laurent's expansion, of a complex function f(z) is defined as a representation of that function in terms of power series that includes the terms of negative degree.
Laurent's series was first published by Pierre Alphonse Laurent in 1843..
What is series expansion method?
In mathematics, a series expansion is a technique that expresses a function as an infinite sum, or series, of simpler functions.
It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division)..
What is the complex function expansion?
Laurent's series, also known as Laurent's expansion, of a complex function f(z) is defined as a representation of that function in terms of power series that includes the terms of negative degree.
Laurent's series was first published by Pierre Alphonse Laurent in 1843..
What is the method of series expansion?
A series expansion is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division).
The resulting series often can be limited to a finite number of terms, thus yielding an approximation of the function..
What is the series expansion of an expression?
A series expansion is a representation of a mathematical expression in terms of one of the variables, often using the derivative of the expression to compute successive terms in the series..
Why do we need series expansion?
In mathematics, a series expansion is a technique that expresses a function as an infinite sum, or series, of simpler functions.
It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division)..
Why is the Laurent series important?
The method of Laurent series expansions is an important tool in complex analysis.
Where a Taylor series can only be used to describe the analytic part of a function, Laurent series allows us to work around the singularities of a complex function..
- A function is complex differentiable if it satisfies the two Cauchy-Riemann equations.
This is important because complex differentiable functions are analytic functions, and analytic functions can be expressed as a Taylor series of a complex variable. - A Maclaurin series is a function that has expansion series that gives the sum of derivatives of that function.
The Maclaurin series of a function. f ( x ) up to order n may be found using Series. [ f , x , 0 , n ] - Show that an=bn for all n∈Z.
This means that the Laurent series expansion is unique.
Hint: It suffices to show that if f≡0, then an=0 for all n.
Use ∑∞n=0anzn=∑−1n=−∞−anzn to construct a bounded entire function. - The method of Laurent series expansions is an important tool in complex analysis.
Where a Taylor series can only be used to describe the analytic part of a function, Laurent series allows us to work around the singularities of a complex function.
