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1: 35.2 Laplace Transform
35.2.1 g ( 𝐙 ) = 𝛀 etr ( 𝐙 𝐗 ) f ( 𝐗 ) d 𝐗 ,
Suppose there exists a constant 𝐗 0 𝛀 such that | f ( 𝐗 ) | < etr ( 𝐗 0 𝐗 ) for all 𝐗 𝛀 . …
2: 35.5 Bessel Functions of Matrix Argument
35.5.3 B ν ( 𝐓 ) = 𝛀 etr ( ( 𝐓 𝐗 + 𝐗 1 ) ) | 𝐗 | ν 1 2 ( m + 1 ) d 𝐗 , ν , 𝐓 𝛀 .
3: 1.1 Special Notation
x , y real variables.
etr ( 𝐀 ) exponential of tr ( 𝐀 )
4: 35.3 Multivariate Gamma and Beta Functions
35.3.2 Γ m ( s 1 , , s m ) = 𝛀 etr ( 𝐗 ) | 𝐗 | s m 1 2 ( m + 1 ) j = 1 m 1 | ( 𝐗 ) j | s j s j + 1 d 𝐗 , s j , ( s j ) > 1 2 ( j 1 ) , j = 1 , , m .
5: 35.6 Confluent Hypergeometric Functions of Matrix Argument
35.6.2 Ψ ( a ; b ; 𝐓 ) = 1 Γ m ( a ) 𝛀 etr ( 𝐓 𝐗 ) | 𝐗 | a 1 2 ( m + 1 ) | 𝐈 + 𝐗 | b a 1 2 ( m + 1 ) d 𝐗 , ( a ) > 1 2 ( m 1 ) , 𝐓 𝛀 .
6: 35.8 Generalized Hypergeometric Functions of Matrix Argument
7: 35.4 Partitions and Zonal Polynomials
8: 35.1 Special Notation
a , b complex variables.
etr ( 𝐗 ) exp ( tr 𝐗 ) .
9: 1.2 Elementary Algebra
The Matrix Exponential and the Exponential of the Trace
1.2.77 det ( exp ( 𝐀 ) ) = exp ( tr ( 𝐀 ) ) = etr ( 𝐀 ) .
10: 21.6 Products
21.6.3 j = 1 h θ ( k = 1 h T j k 𝐳 k | 𝛀 ) = 1 𝒟 g 𝐀 𝒦 𝐁 𝒦 e 2 π i tr [ 1 2 𝐀 T 𝛀 𝐀 + 𝐀 T [ 𝐙 + 𝐁 ] ] j = 1 h θ ( 𝐳 j + 𝛀 𝐚 j + 𝐛 j | 𝛀 ) ,