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from sympy import *
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from convert_formula import *
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from number_list_limitation import limitation
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from sympy import cos, sin
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def mi_grade_number_zone(formula):
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"""
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计算幂级数的收敛域
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:param formula: 幂级数的通项系数
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:return: 收敛域
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"""
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expression = 'abs((' + formula.replace('n', '(n+1)') + ')/' + formula + ')'
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r = limitation(expression)[0]
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if r == 0:
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return '(-oo, oo)'
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else:
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border = [False, False]
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grade_num = '(' + formula + ')*(1/%s)^n'%r
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if limitation(grade_num)[1] == True:
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border[1] = True
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grade_num = '(' + formula + ')*(-1/%s)^n'%r
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if limitation(grade_num)[1] == True:
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border[0] = True
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if border[0] == True and border[1] == True:
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return '['+ str(-1/r) + ', ' + str(1/r) + ']'
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elif border[0] == False and border[1] == False:
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return '('+ str(-1/r) + ', ' + str(1/r) + ')'
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elif border[0] == True and border[1] == False:
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return '('+ str(-1/r) + ', ' + str(1/r) + ']'
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elif border[0] == False and border[1] == True:
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return '['+ str(-1/r) + ', ' + str(1/r) + ')'
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def mi_grade_number_sum(formula):
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"""
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计算幂级数的和函数
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:param formula: 幂级数的通项
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:return: 幂级数的和函数
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"""
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series_expr = convert_formula(formula)
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x = symbols('x')
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series_expr = str(series_expr).replace('n', 'i')
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series_expr = convert_formula(series_expr)
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series_expr = sympify(series_expr)
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i, n = symbols('i n')
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partial_sum = simplify(summation(series_expr, (i, 1, n)))
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if isinstance(partial_sum, Piecewise):
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for expr, condition in partial_sum.args:
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if condition == True:
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partial_sum = expr
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break
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return partial_sum
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def mi_grade_number_series(function, n, x0=0):
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function = convert_formula(function)
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independent_variable, function = split_function(function)
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independent_variable = symbols(independent_variable)
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function = sympify(function)
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taylor_series = function.subs(independent_variable, x0)
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for i in range(1, n+1):
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taylor_series += diff(function, independent_variable, i).subs(independent_variable, x0) * (independent_variable - x0)**i / factorial(i)
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return taylor_series
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def fuliye_grade_number(formula, T=2*pi):
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"""
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把函数展开成傅里叶级数
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param formula: 函数表达式在(-T/2, T/2)上的表达式
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param T: 周期
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:return: 傅里叶级数展开式
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"""
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# 转换公式格式（复用现有工具函数）
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formula = convert_formula(formula)
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# 拆分自变量和函数表达式（假设split_function来自convert_formula.py）
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independent_variable, function = split_function(formula)
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# 符号化自变量和n（n作为符号保留）
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x, t = symbols(f'{independent_variable} t')
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n = symbols('n', integer=True, positive=True)
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# 转换为sympy表达式
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f = sympify(function)
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# 构造傅里叶系数
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a_0 = ((2/t) * integrate(f, (x, -t/2, t/2))).subs(t, T)
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a_n = ((2/t) * integrate(f * cos(2 * n * x * pi / t), (x, -t/2, t/2))).subs(t, T)
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b_n = ((2/t) * integrate(f * sin(2 * n * x * pi / t), (x, -t/2, t/2))).subs(t, T)
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# 构造傅里叶级数
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f_series = a_0/2 + Sum(a_n * cos(2 * n * pi * x / T) + b_n * sin(2 * n * pi * x / T), (n, 1, oo))
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return f_series
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def test():
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"""
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测试上述所有函数的测试函数
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"""
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# 导入必要的库
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# 测试 mi_grade_number_zone 函数
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formula_zone = 'n'
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print("mi_grade_number_zone 测试结果:", mi_grade_number_zone(formula_zone))
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# 测试 mi_grade_number_sum 函数
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formula_sum = 'n*x**n'
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print("mi_grade_number_sum 测试结果:", mi_grade_number_sum(formula_sum))
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# 测试 mi_grade_number_series 函数
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formula_series = 'f(x)=exp(x)'
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print("mi_grade_number_series 测试结果:", mi_grade_number_series(formula_series, 10))
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# 测试 fuliye_grade_number 函数
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formula_fourier = 'f(x)=x'
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print("fuliye_grade_number 测试结果:", fuliye_grade_number(formula_fourier))
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def UI():
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while True:
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print("欢迎使用级数计算器")
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print("1. 计算幂级数的收敛域")
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print("2. 计算幂级数的和函数")
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print("3. 计算幂级数的展开式")
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print("4. 计算傅里叶级数展开式")
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print("5. 退出")
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choice = input("请输入您的选择（1/2/3/4/5）：")
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if choice not in ['1', '2', '3', '4', '5']:
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print("无效选择，请重新输入！")
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continue
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if choice == '1':
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formula_zone = input("请输入幂级数的通项的系数：")
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print("幂级数的收敛域为：", mi_grade_number_zone(formula_zone))
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elif choice == '2':
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formula_sum = input("请输入幂级数的通项：")
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print("幂级数的和函数为：", mi_grade_number_sum(formula_sum))
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elif choice == '3':
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formula_series = input("请输入函数表达式，格式f(x)=...：")
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n = int(input("请输入展开的项数："))
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print("幂级数的展开式为：", mi_grade_number_series(formula_series, n))
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elif choice == '4':
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formula_fourier = input("请输入函数表达式，格式f(x)=...：")
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T = input("请输入周期T，为2π请留空：")
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if T == '':
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T = 2*pi
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else:
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T = float(T)
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print("傅里叶级数展开式为：", fuliye_grade_number(formula_fourier, T))
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elif choice == '5':
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print("谢谢使用，再见！")
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break
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if __name__ == "__main__":
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test()
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UI()
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