What are coherent and incoherent radiation? What are coherent and incoherent radiation?
I am a mathematician who is self-learning physics. In reading Jackson's electrodynamics and other books, I often hear that radiation is incoherent or coherent. What does this mean? Does it just mean that the phases are the same in coherent radiation and are different in incoherent radiation?
 A: Yes, coherent radiation means that the phases of two ( or more ) waves representing the radiation differ by a known constant. 
Incoherence means that the phase differences are unknown/random.
Laser radiation is coherent because stimulated emission assures phase differences are constant . Radiation from an incandescent lamp is incoherent because the electromagnetic waves are generated in a statistically random manner depending on which atoms are excited and de-excited.
To develop intuition think of the classical example of soldiers marching in step. Their motion is coherent and so are the vibrations they set up. They break step over ancient bridges so as to become incoherent, there were cases where ancient bridges resonated to the step and were damaged.
A: Radiation is temporally coherent, if its coherence time is greater than some agreed value, incoherent otherwise (the value is chosen depending on the context). Coherence time is characteristic time of decay of the temporal auto-covariance function of the electric field
$$
\langle E (x,t) E(x,t+\tau)\rangle
$$
as $\tau$ increases.
Analogously, radiation is spatially coherent, if its coherence length is greater than some agreed value. Coherence length is characteristic length of decay of spatial auto-covariance function
$$
\langle E (x,t) E(x+\xi,t)\rangle
$$
as $\xi$ increases.
A: In addition, if the coherence is about spatial and temporal  coherence, and it can be interpreted as $$\Gamma(P_1,P_2;\tau)=\left< E(P_1,t+\tau)E^*(P_2,t)\right>$$
So we call it mutual correlation function, which is important in statistics optics. By the way, if two points with time interval $\tau$ are incoherent, the mutual correlation function is zero.
And if we ignore the spatial coherence or we think these two points coincide, we can get
$$\Gamma(P;\tau)=\left<E(P;t+\tau)E^*(P;t)\right>$$
which is called temporal coherence.
