Real part
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In mathematics, the real part of a complex number z, is the first element of the ordered pair of real numbers representing z, i.e. if z = (x,y), or equivalently, z = x + iy, then the real part of z is x. It is denoted by Re{z} or 
, where 
is a capital R in the Fraktur typeface. The complex function which maps z to the real part of z is not holomorphic.
In terms of the complex conjugate 
, the real part of z is equal to 
.
For a complex number in polar form, z = (r,θ), or equivalently, z = r(cosθ + isinθ), it follows from Euler's formula that z = reiθ, and hence that the real part of reiθ is rcosθ.
Sometimes computations with real periodic functions such as alternating currents and electromagnetic fields are simplified by writing them as the real parts of complex functions. See for example electrical impedance.
