Page 4 - Advanced_Engineering_Mathematics o'neil

2 Guide to Notation

∂( f, g)
Jacobian of f and g with respect to u and v
∂(u,v)

f (x, y, z)dσ surface integral of f over

f (x 0 −), f (x 0 +) left and right limits, respectively, of f (x) at x 0
F[ f ]or f ˆ Fourier transform of f
F[ f ](ω) or F(ω) Fourier transform of f evaluated at ω
ˆ
−1
F inverse Fourier transform
ˆ
F C [ f ]or f C Fourier cosine transform of f
−1 −1
F C or f ˆ C inverse Fourier cosine transform
ˆ
F S [ f ]or f S Fourier sine transform of f
−1 −1
F S or f ˆ S inverse Fourier sine transform
D[u] discrete N - point Fourier transform (DFT) of a sequence u j
ˆ
f win windowed Fourier transform
often denotes the characteristic function of an interval I
χ I
σ N (t) often denotes the Nth Cesàro sum of a Fourier series
Z(t) in the context of filtering, denotes a filter function
P n (x) nth Legendre polynomial
(x) gamma function
B(x, y) beta function
Bessel function of the first kind of order ν
J ν
γ depending on context, may denote Euler’s constant
Bessel function of the second kind of order ν
Y ν
modified Bessel functions of the first and second kinds, respectively, of order zero
I 0 , K 0
2
∇ u Laplacian of u
Re(z) real part of a complex number z
Im(z) imaginary part of a complex number z
z complex conjugate of z
|z| magnitude (also norm or modulus) of z
arg(z) argument of z

f (z)dz integral of a complex function f (z) over a curve C
C
f (z)dz integral of f over a closed curve C
C
Res( f, z 0 ) residue of f (z) at z 0
f : D → D ∗ f is a mapping from D to D ∗

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October 14, 2010 15:48 THM/NEIL Page-2 27410_00_IFC_p01-02