For a complex function with an isolated singularity at a point z0, the residue at z0 is the a-1 coefficient of the function's Laurent series about z0:
       inf
       --                 n
f(z) = >   a_n  (z - z_0)
       --
     n=-inf


Res( f(z); z_0 ) = a_-1

Residues are used with the Residue theorem to calculate contour integrals of functions in the complex plane more easily. The value of an integral over a closed curve C is simply 2*pi*i times the sum of the residues of the singularities within it.

   /                   --
   |  f(z) dz = 2*pi*i >  Res( f(z); z_0 )
   /c                  --
                       z_0

Calculating residues at poles is easy. Suppose f(z) has an nth order pole at z0, i.e., it can be written in the form f(z) = phi(z)/(z-z0)n. Then

                     (n-1)
                   phi    (z_0)
Res( f(z); z_0 ) = ------------
                      (n-1)!

(Superscript here indicating the (n-1)st derivative at z_0).

Example: Find the residue of f(z) = (e^z)/z at z=0.

The Laurent series about the origin is

       1   /         z^2   z^3       \
f(z) = - * | 1 + z + --- + --- + ... |
       z   \          2     3!       /
 
       1       z
     = - + 1 + - + ...
       z       2
So by examination of the first term, the residue is just 1. Therefore any contour integral of f(z) around a closed curve containing the origin will be 2*pi*i.

Res"i*due (r?z"?-d?), n. [F. r'esidu, L. residuum, fr. residuus that is left behind, remaining, fr. residere to remain behind. See Reside, and cf. Residuum.]

1.

That which remains after a part is taken, separated, removed, or designated; remnant; remainder.

The residue of them will I deliver to the sword. Jer. xv. 9.

If church power had then prevailed over its victims, not a residue of English liberty would have been saved. I. Taylor.

2. Law

That part of a testeator's estate wwhich is not disposed of in his will by particular and special legacies and devises, and which remains after payment of debts and legacies.

3. Chem.

That which remains of a molecule after the removal of a portion of its constituents; hence, an atom or group regarded as a portion of a molecule; -- used as nearly equivalent to radical, but in a more general sense.

<-- also moiety -->

⇒ The term radical is sometimes restricted to groups containing carbon, the term residue being applied to the others.

4. Theory of Numbers

Any positive or negative number that differs from a given number by a multiple of a given modulus; thus, if 7 is the modulus, and 9 the given number, the numbers -5, 2, 16, 23, etc., are residues.

Syn. -- Rest; remainder; remnant; balance; residuum; remains; leavings; relics.

 

© Webster 1913.

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