There are many conventions for specifying magnetic fields in terms of a multipole, polynomial, or Taylor expansion, which leads to potential confusion. In elegant (as in MAD[2]), magnet strengths are specified in terms of Taylor series. For normal multipoles and y = 0, the expansion is

| (16) |

where B_{0} is the dipole, B_{1} is the quadrupole, etc. In general,

| (17) |

elegant follows MAD [2] in using a right-handed coordinate system (x,y,z) in which z is along the beam direction, x is to the left, and y is up.

This expansion for the normal multipole terms can be related to a multipole expansion that includes
both normal and skew components. In this convention, positive normal multipole coefficients give positive
B_{y} for x > 0 and y = 0. Rotating a positive normal multipole with N poles π∕N clockwise about the
vector along the beam direction will convert it into a positive skew multipole. As a result, for a positive
skew multipole, B_{y} will be non-negative and B_{x} will be negative for x > 0 along the line
ϕ = π∕N.

We can satisfy these conventions if we write the scalar potential as

| (18) |

where, as we’ll see, A_{m} are skew components and B_{m} are normal components for a 2(m + 1)-pole. The
coordinates (x,y) are in a right-handed system with the longitudinal coordinate z. Δϕ is the rotation
angle of the magnet, where a clockwise rotation about the nominal trajectory corresponds to Δϕ > 0. The
minus sign in e^{-inΔϕ} is because we rotate the magnet while keeping the coordinate system
fixed.

The magnetic fields are

| (19) |

and

| (20) |

We can relate the coefficients to the B_{m} quantities used in MAD and elegant by noting that for
Δϕ = 0

| (21) |

and

| (22) |

Note the minus sign in the last equation, which differs from commonly asserted conventions.

Multipole errors are typically specified as fractions of the main field harmonic at a reference radius R, e.g.,

| (23) |

where m is the main harmonic and n is the error harmonic.

For electrons, the deflection from a thin element is

| (24) |

where H = Bρ = -p∕e is the beam rigidity and p = m_{e}cβγ is the momentum. The geometric strengths
K_{n} are defined as

| (25) |

By convention in elegant, a positive K_{n} value deflects a particle at x > 0 toward x = 0. E.g., a positive
K_{1} value indicates a horizontally focusing quadrupole.