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I would like to perform mathematical functions similar to those you would use Raster Calculator for in ArcMap, however I'd like to do it in Python without using the ArcpPy module. I have the osgeo module, and it seems like it should be possible to do however I'm having a hard time finding any useful syntax.

An example function would be to multiply every cell value in an input raster by 2.

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Without reinventing the wheel, I suggest you to use gdal_calc.py. Example:

gdal_calc.py -A input.tif --outfile=result.tif --calc="A*2"

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