§ 3 Integrals of functions of complex variables
First,
the definition and formula of the integral of the complex variable function
Let be an oriented curve in the complex plane_{}
_{}
_{}The positive direction of is along the direction of parameter increase , and the curve in the opposite direction is denoted as . For any arbitrary_{}_{}
_{}
there is a corresponding
_{}
They divide the curve into segment arcs . If the single value is continuous on the curve , then when_{}_{}_{}_{}
_{}
Japanese
_{}
When there is a limit of , the limit is called the integral of the function along the curve , denoted as_{}_{}
_{}
If it is a piecewise smooth curve * , which is a piecewise continuous and bounded function on , then the function must be integrable along the curve ._{}_{}_{}_{}_{}
The integral of the complex variable function along the curve can actually be transformed into the curve integral of two twovariable real variable functions, so their integral formulas have many similarities (in the following formulas, the integrand functions are assumed to be on a piecebypiece smooth curve . Single value continuous):_{}_{}
_{} ( is a complex constant)_{}
_{}
_{}
( It is made of curves and connections)_{}_{}_{}
_{}
Set on the curve , the length of the curve is , then_{}_{}_{}_{}
_{}
_{ }
Second,
the properties of the integral of the analytic function
[ Cauchy's integral theorem ] The Cauchy integral theorem has the following forms:
1 ^{o} If the function is analytic in a simply connected region , then the integral along any simple closed curve ** in it ( Fig. 10.6 ( a )) is equal to zero, i.e. _{}_{}_{}_{}_{}
_{}
2 ^{o} If the function is resolved in a multiconnected region and is any two closed curves around the same hole (Fig. 10.6 ( b )), then _{}_{}_{}_{}
_{}
3 ^{o} If the function is resolved in a simply connected region , continuous over the package * of , then the integral along the region boundary is equal to zero (Fig. 10.7 ( a )), i.e. _{}_{}_{}_{}_{}
_{}
4 ^{o} If the function is analytic in a multiconnected region , continuous in the upper region, then the integral along the region boundary (Fig. 10.7 ( b )) is equal to zero, i.e. _{}_{}_{}_{}
_{}
[ Indefinite integral ] A function whose derivative is equal to is called an indefinite integral (original function) ._{}_{}
According to Cauchy's theorem, the integral of an analytic function in a simply connected region along any piecewise smooth curve in the region is only related to the two endpoints of the curve and has nothing to do with the route of the integral (Figure 10.8 ), so it can be written as_{}_{}_{}

_{}
Let be any indefinite integral, then_{}_{}
_{}
[ Cauchy integral formula ]
If the function is analyzed in the region D enclosed by a simple closed curve , and is continuous on the above, then for any point in _{D} , we have_{}_{}_{}_{}_{}
_{} ( take the positive direction)_{}
This formula shows that the value of the analytic function at any point in the region can be determined by the value on the boundary, and it also shows that the value of the analytic function in the region has a close relationship with the boundary value, which is of great significance in application ( Figure 10.9 ( a )) .
The Cauchy integral formula also holds for the multiconnected region enclosed by finite simple closed curves (Fig. 10.9 ( b )) .
The Cauchy integral formula also holds for unbounded regions (Fig. 10.9 ( c )): if the bounds of the unbounded region (inclusive ) are finite simple closed curves , the function is analytic except for points inside and continuous and exist at the same time, then for any point within_{}_{}_{}_{}_{}_{}_{}_{}_{}_{}_{}_{}
_{}
_{}the direction is to make the left side of it )_{}
[ Cauchytype integral ] Suppose a closed or nonclosed piecewise smooth curve is a continuous function on , then for any point not on , the integral_{}_{}_{}_{}
_{}
is a singlevalued function, called the Cauchy integral of , denoted as _{.}_{}_{}
The Cauchytype integral is analytic in any simply connected region of points that do not contain a curve , and its higherorder derivative is_{}_{}
_{}
_{} 
[ Average Theorem ] If the function is analyzed in a circle with a center and a radius , and is continuous on the circle, then the value of the function at the center of the circle is equal to the arithmetic mean of the values on the circumference, that is,_{}_{}_{}_{}_{}_{}
_{}
[ Maximum Modulus Theorem ] If the function is analytic in a bounded region , continuous on , and the maximum value set on is , then there is a point on the boundary of , such that for all the_{}_{}_{}_{}_{}_{}_{}_{}_{}_{}_{}_{}
_{}
[ HigherOrder Derivative Theorem ] If the function is analytic in the region and continuous in the upper region, then there are derivatives of various orders at every point within it, and there are_{}_{}_{}_{}_{}
_{} ( yes border)_{}_{}
This theorem shows that as long as it exists, then the higherorder derivative also exists, which is a property that the real variable function generally does not have ._{}
[ Cauchy's inequality ] If the function is analyzed in the area , continuous on the upper, and the shortest distance from the point to the boundary is , and the length is , then_{}_{}_{}_{}_{}_{}_{}_{}_{}_{}
_{}
In particular, when it is a circle , there are inequalities_{}_{}
_{}
[ Liuville's Theorem ] If a function is analytic and bounded in the full plane, then it must be equal to a constant ._{}
[ Morella's Theorem ] If a function is continuous in a simply connected region , and the integral along any simple closed curve in it is equal to zero, then it is analytic in the region ._{}_{}_{}_{}_{}_{}_{}
[ Poisson Formula of Harmonic Function ] Suppose it is harmonic in a circle , continuous on a closed circle , and is any point in the circle, then_{}_{}_{}_{}
_{}
In polar coordinates, there are forms:
_{}
* A curve (or arc) represented by a continuous function, whencontinuous and not equal to zero, and at the same time, this curve is called a smooth curve.The curve composed of finite smooth curves is called a piecewise smooth curve. _{}_{}_{}_{}
* * A continuous curve:twoends of the curve coincide, that is, at the same timeiscalledthe positive direction of the curve is specified as the anticlockwise direction. The unbounded part is calledthe outer. _{}_{}_{}_{}_{}_{}_{}_{}_{}_{}
^{*} Package is defined in Chapter 21 §3, II.