Where z=x+iy, u(x, y) and v(x, y) are real and imaginary parts, and I is imaginary unit (2=? 1i2=？ 1)。 The integration of complex variable function is carried out on the complex plane, which is widely used in mathematical analysis, physics, engineering and other fields.

In the integration of complex variable function, path is the key concept of integration. The path in the complex domain is usually represented by C, which can be a simple curve, a combination of curves or a closed curve. The integration along path c is usually expressed as:

∫Cf(z)dz

Where f(z) is a complex variable function and dz represents a tiny line element on the path. The integral path of a complex number can be a straight line, an arc, a polygon, etc.

Integral types of complex variable functions:

In the complex variable function integration, the common types are:

1. Integration along this road:

This integration means that the complex function \(f(z)\) is integrated along the path \(C\). According to different paths, integration can be divided into forward integration and reverse integration.

2. The contour integral:

Contour integration refers to calculating the integration in the integration path (c) when the integration path is a closed curve. Contour integration is usually expressed by \(\ point _ C f(z)\, dz\), which means that the integration path is closed.

3. Series expansion:

The integral of complex variable function can also be calculated by series expansion, for example, using Laurent series expansion to express complex variable function as power series, so as to integrate.