1, using Taylor expansion: Euler infinite series is an infinite series, which can be expressed as: f (z) = A0+a 1 z+A2Z2+A3Z3++ANZn, where A0, A1,A2 is a constant and Z is a complex number.
If f(z) converges to a certain point z0, then in a certain neighborhood of z0, f(z) can be approximately expressed as Taylor expansion: f(z)=f(z0)+f'(z0)(z? z0)+f ”( z0)(z？ z0)22！ +f？ (z0)(z？ z0)33！ +,where f'(z0), f "(z0), f? (z0) is the derivative of f(z) at z0. By comparing the coefficients, the summation formula of Euler infinite series can be obtained.
2. Using power series expansion: Euler infinite series can also be expressed as power series expansion: f(z)=∑n=0∞anzn where an is a constant and z is a complex number. Through the expansion of power series, the summation formula of Euler infinite series can be obtained.
3. Using differential equations: Euler infinite series can also be solved by differential equations. Let f(z) converge to a certain point z0 and be derivable in a certain neighborhood of z0, then the differential equation can be established: f ′ (z) = a1+2a2z+3a3z2++(n? 1) Ann? 1zn？ 2+nanzn？ 1。 By solving the differential equation, the summation formula of Euler infinite series can be obtained.
Application of Euler infinite series;
1. Solving differential equations: Euler infinite series can be used to solve some differential equations. For example, Euler formula can be expressed as infinite series, which is very useful in solving some differential equations. By transforming the differential equation into infinite series, the calculation can be simplified and the exact solution can be obtained.
2. Approximate calculation: Euler infinite series can be used to approximate the values of some functions. For example, when calculating the approximate value of pi, Euler infinite series expansion can be used to calculate it. Although this method is not the most accurate, it is still an effective approximate calculation method.
3. Properties of analytic functions: Euler infinite series can be used to study the properties of analytic functions. For example, through Euler formula, we can study the properties of functions in complex fields. Euler infinite series can also be used to solve the integral and power series expansion of some special functions. Euler infinite series is widely used in mathematics and physics, not only in mathematical analysis, differential equations, complex variable functions and other fields, but also in physics and engineering.