T.H.
Welcome to Los Angeles :)
The problem with Burbank is the hospital, St. Joseph's, is NOT a birth friendly hospital so most of the good doctors don't/won't work there. There is one good doctor there:
Christopher Pearson
1411 W Olive, Ave D
Burbank, CA 91506
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If you birth at St. Joe's, you'll want to stay home as long as possible - they won't 'let you' get out of bed once you're in active labor or your bag of waters has released ... and being stuck in bed makes it very hard to labor well! There's a great nurse there named Susan Chait -- ask for her if you're there and she'll support your birth plan :)
Happy Birthing!