1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5.4 • TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 179

where coshy = e• + 2 • - • and sinhy = ••7-•, respectively, are the hyperbolic
cosine and hyperbolic sine functions that you studied in calculus.

Similarly,

cosz=^1.. )
2

(e"+e- u

= ~ ( ei(x+iy) + e -i(x+iy))

=^1 2 (e- Y +· t:Z: + eY-•X. )

= ~ (e-Y (cosx + isinx) +ell (cosx - isinx)]

= cos x (ell~ e-Y) -i sin x ( eY -2 e-Y)

= cos x cosh y - isin x sinh y.

(5- 34 )

(5-35)

Equipped with Identities (5-32)-(5-35), we can now establish many other
properties of the trigonometric functions. We begjn with some periodic results.

• For all complex numbers z = x + i y ,

sin (z + 2tr) = sinz,

cos (z + 2tr) = cosz,

sin (z +tr)= -sinz,

cos(z+ tr) = - cosz,

tan(z+ tr) = tanz, and

cot ( z + 7r) = cot z.

Clearly, sin (z + 2tr) = sin !(x + 2tr) + iy]. By Identity (5-33) this expression

is sin (x + 2tr) cosh y + icos (x + 2tr) sinh y = sinx cosh y + i cosx sinh y =

sin z. Again, the proofs for the other periodic results are left as exercises.

• If z1 and z 2 are any complex numbers, then

sin (z1 + zi) = sin z1 cos z2 + cos z1 sin z2 and

cos~+Zz)=cosz1coszi-~~~Zz. w

sin2z = 2sinzcosz,

cos 2z = cos^2 z -sin^2 z , and

sin (i +z) =sin(~ -z) = cosz.