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Goodwill and Other Identifiable Intangible Assets, Net (Details) - USD ($)
$ in Thousands
 3 Months Ended 6 Months Ended
Mar. 31, 2023
Mar. 31, 2024
Mar. 31, 2023
Sep. 30, 2023
Finite-Lived Intangible Assets [Line Items]        
Gross carrying amount   $ 436,063   $ 435,897
Accumulated amortization   (171,180)   (158,813)
Net   264,883   277,084
Amortization expense   12,754 $ 13,702  
Goodwill   342,022   341,593
Purchased and Core Technology        
Finite-Lived Intangible Assets [Line Items]        
Gross carrying amount   85,038   85,032
Accumulated amortization   (61,745)   (59,833)
Net   23,293   25,199
License Agreements        
Finite-Lived Intangible Assets [Line Items]        
Gross carrying amount   112   112
Accumulated amortization   (112)   (112)
Net   0   0
Patents and Trademarks        
Finite-Lived Intangible Assets [Line Items]        
Gross carrying amount   40,091   39,957
Accumulated amortization   (20,953)   (19,888)
Net   19,138   20,069
Customer Relationships        
Finite-Lived Intangible Assets [Line Items]        
Gross carrying amount   309,222   309,196
Accumulated amortization   (86,770)   (77,380)
Net   222,452   231,816
Non-compete Agreements        
Finite-Lived Intangible Assets [Line Items]        
Gross carrying amount   600   600
Accumulated amortization   (600)   (600)
Net   0   0
Order backlog        
Finite-Lived Intangible Assets [Line Items]        
Gross carrying amount   1,000   1,000
Accumulated amortization   (1,000)   (1,000)
Net   0   $ 0
Cost of Sales and General and Administrative Expense        
Finite-Lived Intangible Assets [Line Items]        
Amortization expense $ 6,100 $ 12,300 $ 12,700