Determinants and Their Applications in Mathematical Physics

84 4. Particular Determinants

the simple hyperbolic functions;

A(H 1 ,H 2 )=

∣

∣

∣

∣

H 1 H 2

H 2 H 1

∣

∣

∣

∣

=1. (4.4.19)

Whenn=3,ωr= exp(2riπ/3),

ω

3

r=1,

ω=ω 1 = exp(2iπ/3),

ω

2

= ̄ω,

ωω ̄=1.

W=





11 1

1 ωω

2

1 ω

2

ω





,

W

− 1

=

1

3

W,

Er= exp

[

2

∑

t=0

ω

(r−1)t

xt

]

= exp

[

ω

r− 1

x 1 +ω

2 r− 2

x 2

]

.

Let (x 1 ,x 2 )→(x, y). Then,

E 1 = exp(x+y),

E 2 = exp(ωx+ ̄ωy),

E 3 = exp( ̄ωx+ωy)=E ̄ 2. (4.4.20)





H 1

H 2

H 3



=^1

3





111

1 ω ω ̄

1 ̄ωω









E 1

E 2

E 3



, (4.4.21)

H 1 =

1

3

[

e

x+y

+e

ωx+ ̄ωy

+e

ωx ̄+ωy

]

,

H 2 =

1

3

[

e

x+y

+ωe

ωx+ ̄ωy

+ ̄ωe

ωx ̄+ωy

]

,

H 3 =

1

3

[

e

x+y

+ ̄ωe

ωx+ ̄ωy

+ωe

ωx ̄+ωy

]

. (4.4.22)

Since the complex terms appear in conjugate pairs, all three functions are

real:

A(H 1 ,H 2 ,H 3 )=

∣ ∣ ∣ ∣ ∣ ∣

H 1 H 2 H 3

H 3 H 1 H 2

H 2 H 3 H 1

∣ ∣ ∣ ∣ ∣ ∣

=1. (4.4.23)

A bibliography covering the years 1757–1955 on higher-order sine func-

tions, which are closely related to higher-order or generalized hyperbolic

functions, is given by Kaufman. Further notes on the subject are given by

Schmidt and Pipes, who refer to the generalized hyperbolic functions as

cyclical functions and by Izvercianu and Vein who refer to the generalized

hyperbolic functions as Appell functions.