1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

178 CHAPTER, 5 • ELEMENTARY FUNCTIONS

• For all complex numbers z for which the expressions are defined,

d

dz tan z = sec

2

z,

d 2

• cot z = - csc z ,
  dz
  d
  dzsecz = secztanz,
  d
  dz cscz = - csczcotz.

and

The proof that Jz tanz = sec^2 z uses the identity sin^2 z + cos^2 z = 1:

d

• tanz =
  dz

=

!£ (sinz ) = cosz1zsinz-sinz~cosz

dz cos z cos^2 z

cos^2 z+sin^2 z 1

cos^2 z

= sec^2 z.

= cos^2 z

We leave the proofs of the other derivative formulas as exercises.

To establish additional properties, expressing cos z and sin z in the Cartesian

form u +iv will be useful. (Additionally, the applications in Chapters 10 and

11 will use these formulas.) We begin by observing that the argument given to

prove part (iii) in Theorem 5.1 easily generalizes to the complex case with the

aid of Definition 5.5. That is,

ei• = cosz + isinz,

for all z , whether z is real or complex. Hence

e-iz = cos (-z) + isin (-z) = cosz -isinz.

(5-30)

(5-31)

Subtracting Equation (5-31) from Equation (5-30) and solving for sin z give

sinz =^1.. )
2

i (e" - e - t z

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

= -1 ( e -y+ix -e y -ix)

2i

= ~ [e-Y (cosx + isinx) - eY (cosx - isinx)]

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

= sinxcoshy + i cosxsinhy,

(5-32)

(5-33)