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I study the location of the $\gamma$-ray emission in blazar jets by creatinga Compton-scattering approximation valid for all anisotropic radiation fieldsin the Thomson through Klein-Nishina regimes, which is highly accurate and canspeed up numerical calculations by up to a factor $\sim10$. I apply thisapproximation to synchrotron self-Compton, and external Compton-scattering ofphotons from the accretion disk, broad-line region (BLR), and dust torus. I usea stratified BLR model and include detailed Compton-scattering calculations ofa spherical and flattened BLR. I create two dust torus models, one where thetorus is an annulus, and one where it is an extended disk. I present detailedcalculations of the photoabsorption optical depth using my detailed BLR anddust torus models, including the full angle dependence. I apply thesecalculations to the emission from a relativistically moving blob travelingthrough these radiation fields. The ratio of $\gamma$-ray to optical fluxproduces a predictable pattern that could help locate the $\gamma$-ray emissionregion. I show that the bright flare from 3C 454.3 in 2010 November detected bythe Fermi Large Area Telescope is unlikely to originate from a single blobinside the BLR since it moves outside the BLR in a time shorter than the flareduration, although emission by multiple blobs inside the BLR is possible; and$\gamma$-rays are unlikely to originate from outside the BLR from scattering ofphotons from an extended dust torus, since then the cooling timescale would betoo long to explain the observed short variability.Comment: Accepted by ApJ. 22 pages, 19 Figures, 5 Table

EXTERNAL COMPTON SCATTERING IN BLAZAR JETS AND THE LOCATION OF THE GAMMA-RAY EMITTING REGION