University Mathematics Handbook (2015)

XII. Complex Functions

Chapter 3. Elementary Functions

3.1  Exponential Functions, Hyperbolic Functions

a.  For every complex ,

.

If is real , we get an expansion of real function to a power series (see VI,4.7).

b.  For every and there holds .

c.  The equation has no (complex) solutions, that is, has no zeroes.

d.  Hyperbolic sine: (see II.2.8, 2.9)

e.  Hyperbolic cosine: cosh=

(see II.2.8, 2.5)

f.  Hyperbolic tangent: tanh =

g.   has an infinite number of solutions:

h.   has an infinite number of solutions: .

3.2  Trigonometric Functions

For every :

a.  

b.  

c.  The equation has real solutions only:

d.  The equation , has real solutions only:

3.3  Euler's Formula

a.  For every complex ,

b.  ,

c.   is a periodic function, with a period of (see II. 1.4).

d.  Functions and are periodic, with a period of .

e.   for every of .

3.4  Logarithmic Function

a.  A (natural) logarithm of complex number is a complex number holding . It is denoted .

The natural logarithm of real number is usually denoted as .

b.  If , then

or    

c.   is a multivalued function.

d.  For every different from zero, and for every real there holds:

1.  

2.  

3.  

4.  

3.5  Inverse Trigonometric and Hyperbolic Functions

a.   is a complex number holding

b.   is a complex number holding

c.  

d.  

e.  

f.