The notch in a falcon's beak is called the tomial tooth, and it's an adaptation suited to their hunting style.
Here you can see how the top notch (the tomial tooth) and the bottom notch fit together. The powerful beak and shape of the notch work together to allow falcons to bite through the cervical vertebrae and sever the spinal cord of their prey with ease (with Peregrine Falcons, the prey is primarily birds).
The Tomial Tooth is an adaptation unique to falcons. The only other bird that has a tomial tooth is the shrike (click here to learn more about shrikes).
...an entry from my sketchbook on the Peregrine Falcon's tomial tooth.
If you want to accurately represent a falcon in a painting or drawing, it's important to pay attention to the shape of its beak. I wanted to spend a little time studying its unique shape, so I sketched it out a few times.
Also specific to the falcons is the shape of their wings. Falcons get their name from the Latin word "falco" or "falx," which means "sickle" or "scimitar-shaped" and refers to the shape of their wings, which are long and narrow and pointed at the end, similar in shape to a sickle.
...another sketchbook entry. By looking at the silhouette of a falcon, you can see the sickle shape of the wings and their pointed tips. When I first started looking up at the sky to identify raptors by silhouette, it took me a while to figure out the basic forms. After a while, it became clear...the shape of the tail feathers helped me the most. Accipiters (Cooper's Hawks, Sharp-shinned hawks) longer and thinner...and Buteos (Red-tailed Hawks, Red-shouldere Hawks) shorter and wider. Falcons are distinct...fast and sleek with a very thin tail.
Additionally, falcons do not have well-defined supraorbital ridges like hawks and eagles (click here for a few posts that describe the supraorbital ridge--the bony ridge above the eye that helps to cut the sun's glare). Even though falcons have a much less defined bony ridge, they all have a dark stripe under their eye (called a malar stripe), which cuts the sun's glare to help them hunt.
p.s. The close-up shots of the Peregrine Falcon came from the photo-shoot at RAPTOR, Inc. back in autumn. For more close-ups of the beautiful raptors Matty and I photographed that day, click here. As always, artists can feel free to use these photos as reference shots. It's so hard to get close-up references of falcons in the wild. Going to a RAPTOR, Inc. event lets you study the birds and learn their subtleties. I would never have been able to study the notch in the beak if I had not seen this bird up close.