1. The formula connecting complex exponential function and trigonometric function, where E is the base of natural logarithm and I is imaginary unit. It extends the definition of exponential function to complex number and establishes the relationship between trigonometric function and exponential function. It not only appears in mathematical analysis, but also occupies a very important position in the theory of complex variable functions, and is also called "overpass in mathematics".
Second, the proof of Euler formula in the theory of complex variable function;
1, when R=2, these two regions can be imagined as two hemispheres with the equator as the boundary. There are two "vertices" on the equator that divide the equator into two "boundaries", that is, R=2, V=2 and E=2, so R+V-E=2, and euler theorem holds.
2. Let R=m(m≥2) hold for euler theorem. It is proved that euler theorem holds for R=m+ 1. It shows that if we choose an area X on the map with R=m+ 1, then X must have an area Y adjacent to it, so after removing the unique boundary between X and Y, there are only m areas on the map.
3. After removing the boundaries of X and Y, if the vertices at the two ends of the original boundary are still vertices of three or more boundaries, the vertices will remain and the number of other boundaries will remain unchanged; If the vertex at one or both ends of the original boundary is now the vertex of two boundaries, the vertex is deleted, and the two boundaries on both sides of the vertex become one boundary.