Acquisitions (Narrative) (Details) - USD ($)						12 Months Ended
	Jul. 21, 2020	Sep. 03, 2019	Aug. 02, 2019	Apr. 01, 2019	Oct. 01, 2018	Sep. 30, 2020	Sep. 30, 2019	Sep. 30, 2018
Business Acquisition [Line Items]
Gain (Loss) on Disposition of Business								$ 24,500,000
Proceeds from Sales of Business, Affiliate and Productive Assets			$ 166,600,000			$ 800,000	$ 166,600,000	1,000,000.0
Gain (Loss) on Disposition of Other Assets						4,200,000	15,900,000
Cash consideration					$ 17,100,000
Cash payment, net of cash acquired						28,400,000	(303,400,000)	0
Goodwill						1,835,500,000	1,800,900,000	$ 1,738,300,000
Breathe [Member]
Business Acquisition [Line Items]
Cash consideration		$ 127,600,000
Cash payment, net of cash acquired		127,600,000
Goodwill		59,800,000
Business Combination, Recognized Identifiable Assets Acquired and Liabilities Assumed, Property, Plant, and Equipment		$ 2,100,000
Voalte [Member]
Business Acquisition [Line Items]
Business Combination, Contingent Consideration Arrangements, Change in Amount of Contingent Consideration, Liability						$ (8,400,000)	$ 3,200,000
Cash consideration				$ 175,800,000
Cash payment, net of cash acquired				181,000,000.0
Goodwill				98,500,000
Business Combination, Contingent Consideration, Liability				5,200,000
Business Combination, Contingent Consideration Arrangements, Range of Outcomes, Value, High				15,000,000.0
Business Combination, Recognized Identifiable Assets Acquired and Liabilities Assumed, Property, Plant, and Equipment				$ 200,000
Videomed
Business Acquisition [Line Items]
Cash consideration	$ 7,800,000
Cash payment, net of cash acquired	10,700,000
Goodwill	10,000,000.0
Business Combination, Contingent Consideration, Liability	2,900,000
Business Combination, Contingent Consideration Arrangements, Range of Outcomes, Value, High	3,700,000
Business Combination, Step Acquisition, Equity Interest in Acquiree, Remeasurement Gain	$ 3,000,000.0