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.