Laurent ExpansionDate: 02/25/99 at 01:03:03 From: Genni J. & Chris N. Subject: Laurent Expansions Could you explain to us exactly what a Laurent Expansion is and give a few examples? We would appreciate if you could mention all the details. Thank you immensely, Genni and Chris Date: 02/25/99 at 09:35:53 From: Doctor Rob Subject: Re: Laurent Expansions A Laurent series is an expression of this form: a(n)*x^n + a(n-1)*x^(n-1) + ... + a(0) + a(-1)/x + a(-2)/x^2 + ..., where the a(i)'s are constants. It is an infinite sum, consisting of only finite non-zero terms with positive exponents of x, and potentially infinite non-zero terms with negative exponents of x. The constant coefficients can be from any field, and are often the rational numbers, the real numbers, or the complex numbers. These are analogues of decimal expansions of real numbers, which have finitely many digits to the left of the decimal point, and potentially infinitely many nonzero digits to the right of it. As an example one can find the Laurent series expansion of any rational function (quotient of two polynomials) by long division: f(x) = (x^3-2*x)/(x+3). x^2 - 3*x + 7 - 21/x + 63/x^2 - 189/x^2 + ... ------------------------------------------------------- x + 3 ) x^3 - 2*x x^3 + 3*x^2 ----------------- -3*x^2 - 2*x -3*x^2 - 9*x ----------------- 7*x 7*x + 21 ---------------- -21 -21 - 63/x ------------------- 63/x 63/x + 189/x^2 -------------------- -189/x^2 ... So, (x^3-2*x)/(x+3) = x^2 - 3*x + 7 - 21/x + 63/x^2 - 189/x^2 + ... Usually these series are considered as formal objects; that is, no convergence properties have to be taken into account, although one does encounter convergence properties at some point when applying them to physical situations. These series can be manipulated in the same way that polynomials are manipulated, i.e. adding two Laurent series being done term-by-term, and the same for subtracting. Multiplication is done using the Distributive Law, and division using long division as is done above. The polynomial part of a Laurent series is the sum of all the terms with non-negative exponents of x. This is the analogue of the integer part of a decimal real number. In the above example, the polynomial part is x^2 - 3*x + 7. The idea of a degree of a Laurent series makes sense, being the largest exponent of x in any term. In the above example, the degree of f(x) is 2. Such degrees can be negative. The above Laurent series are expansions about x = 0. There also exist Laurent expansions about x = c, for any constant c. These take the form a(n)*(x-c)^n + ... + a(0) + a(-1)/(x-c) + a(-2)/(x-c)^2 + ... You get this from the form above by substituting x-c for x everywhere. A Laurent series expansion of a function about c will not make sense unless there is an integer n (which is the degree) such that the limit as x grows without bound of f(x)/(x-c)^n equals a nonzero constant (which is just a(n)). For example, f(x) = sqrt(x) will not have a sensible Laurent series expansion about 0, since sqrt(x)/x^n has limit 0 if n >= 1, and has limit infinity if n <= 0. Not all Laurent series represent rational functions. It turns out that the rational functions are the ones whose Laurent series coefficients satisfy a linear recurrence relation. Here is another example: x*sin(1/x) = 1 - (1/3!)/x^2 + (1/5!)/x^4 - (1/7!)/x^6 + ... This is definitely not a rational function. It does have a Laurent series expansion about 0 of degree 0. If you have more specific questions, please write back! - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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