11 Basic functions of a complex variable
Recall the definition of the complex exponent - . Then
Maclaurin series expansion. The radius of convergence of this series is +∞, which means that the complex exponent is analytic on the entire complex plane and
(exp z)"=exp z; exp 0=1. (2)
The first equality here follows, for example, from the theorem on term-by-term differentiation of a power series.
11.1 Trigonometric and hyperbolic functions
The sine of a complex variable called a function
Cosine of a complex variable there is a function
Hyperbolic sine of a complex variable is defined like this:
Hyperbolic cosine of a complex variable-- is a function
We note some properties of the newly introduced functions.
A. If x∈ ℝ , then cos x, sin x, ch x, sh x∈ ℝ .
B. There is the following connection between trigonometric and hyperbolic functions:
cos iz=ch z; sin iz=ish z, ch iz=cos z; shiz=isinz.
B. Basic trigonometric and hyperbolic identities:
cos 2 z+sin 2 z=1; ch 2 z-sh 2 z=1.
Proof of the basic hyperbolic identity.
Main trigonometric identity follows from the Ononian hyperbolic identity when the connection between trigonometric and hyperbolic functions is taken into account (see property B)
G Addition Formulas:
In particular,
D. To calculate the derivatives of trigonometric and hyperbolic functions, one should apply the theorem on term-by-term differentiation of a power series. We get:
(cos z)"=-sin z; (sin z)"=cos z; (ch z)"=sh z; (sh z)"=ch z.
E. The functions cos z, ch z are even, while the functions sin z, sh z are odd.
G. (Periodicity) The function e z is periodic with period 2π i. The functions cos z, sin z are periodic with a period of 2π, and the functions ch z, sh z are periodic with a period of 2πi. Furthermore,
Applying the sum formulas, we get
W. Decompositions into real and imaginary parts:
If a single-valued analytic function f(z) bijectively maps a domain D onto a domain G, then D is called a domain of univalence.
AND. Domain D k =( x+iy | 2π k≤ y<2π (k+1)} для любого целого k является областью однолистности функции e z , которая отображает ее на область ℂ* .
Proof. Relation (5) implies that the mapping exp:D k → ℂ is injective. Let w be any nonzero complex number. Then, solving the equations e x =|w| and e iy =w/|w| with real variables x and y (we choose y from the half-interval); sometimes brought into consideration ... ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron
Functions inverse to hyperbolic functions (See Hyperbolic functions) sh x, ch x, th x; they are expressed by formulas (read: hyperbolic aresine, hyperbolic area cosine, aretangent ... ... Great Soviet Encyclopedia
Functions inverse to hyperbolic. functions; expressed in formulas... Natural science. encyclopedic Dictionary
Inverse hyperbolic functions are defined as the inverses of hyperbolic functions. These functions determine the area of the unit hyperbola sector x2 − y2 = 1 in the same way that inverse trigonometric functions determine the length ... ... Wikipedia
Books
- Hyperbolic functions , Yanpolsky A.R. The book describes the properties of hyperbolic and inverse hyperbolic functions and gives the relationship between them and other elementary functions. Applications of hyperbolic functions to…
It can be written in a parametric form using hyperbolic functions (this explains their name).
Denote y= b·sht , then x2 / a2=1+sh2t =ch2t . Whence x=± a·cht .
Thus, we arrive at the following parametric equations of the hyperbola:
Y= in sht , –< t < . (6)
Rice. one.
The "+" sign in the upper formula (6) corresponds to the right branch of the hyperbola, and the ""– "" sign corresponds to the left branch (see Fig. 1). The vertices of the hyperbola A(– a; 0) and B(a; 0) correspond to the value of the parameter t=0.
For comparison, we can give the parametric equations of an ellipse using trigonometric functions:
X=a cost ,
Y=in sint , 0 t 2p . (7)
3. Obviously, the function y=chx is even and takes only positive values. The function y=shx is odd, because :
The functions y=thx and y=cthx are odd as quotients of an even and an odd function. Note that unlike trigonometric functions, hyperbolic functions are not periodic.
4.
Let us study the behavior of the function y= cthx in the neighborhood of the discontinuity point x=0: 
Thus the y-axis is the vertical asymptote of the graph of the function y=cthx . Let us define oblique (horizontal) asymptotes:
Therefore, the line y=1 is the right horizontal asymptote of the graph of the function y=cthx . Due to the oddness of this function, its left horizontal asymptote is the straight line y= –1. It is easy to show that these lines are simultaneously asymptotes for the function y=thx. The functions shx and chx have no asymptotes.
2) (chx)"=shx (displayed similarly).
4) 
There is also a certain analogy with trigonometric functions. A complete table of derivatives of all hyperbolic functions is given in section IV.
Tangent, cotangent
Definitions of hyperbolic functions, their domains of definitions and values
sh x- hyperbolic sine, -∞ < x < +∞; -∞ < y < +∞ .
ch x- hyperbolic cosine
, -∞ < x < +∞; 1 ≤ y< +∞ .
th x- hyperbolic tangent
, -∞ < x < +∞; - 1 < y < +1 .
cth x- hyperbolic cotangent
, x ≠ 0; y< -1 или y > +1 .
Graphs of hyperbolic functions
Plot of the hyperbolic sine y = sh x
Plot of the hyperbolic cosine y = ch x
Plot of the hyperbolic tangent y= th x
Plot of the hyperbolic cotangent y= cth x
Formulas with hyperbolic functions
Relationship with trigonometric functions
sin iz = i sh z ; cos iz = ch z
sh iz = i sin z ; ch iz = cos z
tgiz = i th z ; ctg iz = - i cth z
th iz = i tg z ; cth iz = - i ctg z
Here i is an imaginary unit, i 2 = - 1
.
Applying these formulas to trigonometric functions, we obtain formulas relating hyperbolic functions.
Parity
sh(-x) = - sh x;
ch(-x) = ch x.
th(-x) = -th x;
cth(-x) = - cth x.
Function ch(x)- even. Functions sh(x), th(x), cth(x)- odd.
Difference of squares
ch 2 x - sh 2 x = 1.
Formulas for sum and difference of arguments
sh(x y) = sh x ch y ch x sh y,
ch(x y) = ch x ch y sh x sh y,
,
,
sh 2 x = 2 sh x ch x,
ch 2 x = ch 2 x + sh 2 x = 2 ch 2 x - 1 = 1 + 2 sh 2 x,
.
Formulas for products of hyperbolic sine and cosine
,
,
,
,
,
.
Formulas for the sum and difference of hyperbolic functions
,
,
,
,
.
Relation of hyperbolic sine and cosine with tangent and cotangent
,
,
,
.
Derivatives
,
Integrals of sh x, ch x, th x, cth x
,
,
.
Expansions into series
Inverse functions
Areasine
At - ∞< x < ∞
и - ∞ < y < ∞
имеют место формулы:
,
.
Areacosine
At 1 ≤ x< ∞
and 0 ≤ y< ∞
there are formulas:
,
.
The second branch of the areacosine is located at 1 ≤ x< ∞
and - ∞< y ≤ 0
:
.
Areatangent
At - 1
< x < 1
and - ∞< y < ∞
имеют место формулы:
,
Along with the connection between trigonometric and exponential functions that we discovered in the complex domain (Euler formulas)
in the complex domain there is a very simple connection between trigonometric and hyperbolic functions.
Recall that, according to the definition:
If in identity (3) we replace with then on the right side we get the same expression that is on the right side of the identity, from which the equality of the left sides follows. The same holds for identities (4) and (2).
By dividing both parts of identity (6) into the corresponding parts of identity (5) and vice versa (5) by (6), we obtain:
A similar replacement in identities (1) and (2) and a comparison with identities (3) and (4) gives:
Finally, from identities (9) and (10) we find:
If we put in identities (5) - (12) where x is a real number, i.e., consider the argument purely imaginary, then we get eight more identities between the trigonometric functions of a purely imaginary argument and the corresponding hyperbolic functions of a real argument, as well as between hyperbolic functions of a purely imaginary imaginary Argument and the corresponding trigonometric functions of the real argument:
The relations obtained make it possible to pass from trigonometric functions to hyperbolic ones and from
hyperbolic functions to trigonometric ones with the replacement of the imaginary argument by the real one. They can be formulated as the following rule:
To move from trigonometric functions of an imaginary argument to hyperbolic ones or, conversely, from hyperbolic functions of an imaginary argument to trigonometric ones, one should take the imaginary unit out of the sign of the function for the sine and tangent, and discard it altogether for the cosine.
The connection established is remarkable, in particular, in that it makes it possible to obtain all relations between hyperbolic functions from known relations between trigonometric functions by replacing the latter by hyperbolic functions
Let's show how it is. is being done.
Take for example the basic trigonometric identity
and put in it where x is a real number; we get:
If in this identity we replace the sine and cosine by the hyperbolic sine and cosine according to the formulas, then we get or and this is the basic identity between the previously derived in a different way.
Similarly, you can derive all other formulas, including formulas for the hyperbolic functions of the sum and difference of arguments, double and half arguments, etc., thus, from ordinary trigonometry, get "hyperbolic trigonometry".
