Exponential, hyperbolic and trigonometric functions

Question 1

Exponential series

Answer 1

exp(z) = SUM (n=0, inf ) z^n / n!
for all z e (C

Suppose that z=x+iy, where x,y e |R. Then
exp(z) = e^x(cos(y) + i sin(y))

Question 2

Periodicity of exponential function

Answer 2

exp(z) = exp ( z + 2Pi* i *k )

Question 3

Principal branch of the complex logarithm

Answer 3

The principal branch of the complex logarithm is the function Log from (C \ {0} to (C given by log(z) = log|z| + i Arg(z)

(the same function can be defined with arg(z) however, this can take multiple values so we use the above to above ambiguity)

Question 4

Differential of the principal branch

Answer 4

The principal branch of the complex logarithm Log is differentiable in w e (C \ (- inf, 0] and
Log ‘ (w) = 1 / w for all w e (C \ (-inf, 0].

Question 5

Complex powers of complex numbers

Answer 5

Given z e (C \ {0} and a e (C, we define

z^a = exp (a log(z)).

Question 6

Hyperbolic cosine

Answer 6

cosh(z) =( e^z + e^(-z) ) / 2

= SUM (n e |N, n even) z^n / n!

Question 7

Hyperbolic sine

Answer 7

sinh(z) = ( e^z - e ^(-z) ) / 2

= SUM (n e |N, n odd) z^n / n!

Question 8

Properties of hyperbolic funtions

Answer 8

(1) cosh(-z) = cosh(z)
(2) sinh(-z) = - sinh(z)
(3) cosh’(z) = sinh’(z)
(4) sinh’(z) = cosh(z)
(5) cosh ( z + 2* Pi * k) = cosh (z)
(6) sinh ( z + 2Pik) = sinh(z)

Question 9

Hyperbolic angle sum identities

Answer 9

(1) cosh(z + w) = cosh(z) cosh(w) + sinh(z) sinh(w)
(2) sinh (z + w) = sinh(z)cosh(w) + cosh(z)sinh(w)
(3) cosh^2(z) - sinh^2(z) =1

Question 10

Inverse cosh(z)

Answer 10

We define the inverse function cosh^-1 by
Cosh^-1(w) = Log(w + sqrt(w^2 -1))
for all w e (C.

Question 11

Cos(z) formula

Answer 11

cos(z) =( e^(iz) + e^(-iz) ) / 2

Question 12

sin(z) formula

Answer 12

sin(z) = (e^(iz) - e^(-iz) ) / 2i