It depends on the diameter of the hole (i suspect 7/8" or so hole for 3/4" copper pipe) and it depends on what other studs are in proximity. Since the king/jack of the passage-way door are in a corner, there are other studs within 0-5" of the king/jack, most likely two others, possibly three others.
As @BMitch comments, a 1" hole across the width of a 2-by removes 2/3's of the wood, and is a significant structural change ... to that stud.
Nearly all species of framing lumber have a 'parallel' compressive strength significantly greater than 5000 PSI. After drilling a 1" hole in a 2x4, the narrowest cross section will be 3.5" x 0.5" = 1.75 square inches. As a first estimate, such a 2x4 could independently directly support a 1.75 x 5000 = 8000 pound load, or an 8-ton loaded header, which is equal to the dead+live full load of at least 160 sqft of a 2nd story plus roof. However, the second floor and roof loads are typically split between 2 parallel walls, so the figure is more akin to 320 sqft of a 2nd story plus roof.
That being said, I don't think the compressive strength is a linear function of area and I am not sure the intended valid range of cross-sections for the data in this table that is based on U.S. Forest Products Laboratory data:
Yet even if the 5000 psi figure is meant for 8x8 posts, and is off by a factor of 10 when applied to cross-sections on the order of 3.5x0.5, the drilled out jack stud could still support 8 200 lb people standing on top of the header. There is not even enough room for 8 pairs of feet on 36" header.
So while 1" hole in a 2-by takes away 2/3rds of it's compressive strength, the compressive strength of framing lumber is so great that the resulting 2-by is still overly sufficient as a jack stud.
In the OP's situation, due to the fact that king/jack are in a corner, the wall is significantly over-built from a structural load-bearing perspective.
In fact, from a structural load-bearing perspective, nearly all corners in general are significantly over-built because they are framed with one or more extra studs that are there merely to provide a backing for the edge of sheetrock. These extra studs are not needed to satisfy load bearing requirements.
So, from the structural viewpoint of the wall, a 1" hole is a non-issue in the OP's situation. Go ahead and drill it, preferably with half the hole in the king stud, and the other half in either the corner stud. If the king is serving as a corner stud, then half in the king and half in the jack. Or half in the jack and half in void between the jack and the door jamb (temporarily fill the void to drill).
From a code viewpoint, the inspector might understand this, or might not. Ask him/her before you drill.