For any real number φ, Euler's formula states that the complex exponential function satisfies
Since the multiplicative property still works for the complex exponential function, ez = ez+2πki, for all complex z and integers k.
Since the multiplicative property still works for the complex exponential function, ez = ez+2nπi, for all complex z and integers n.
Alte rezultate
Collapse in complex systems is an exponential function of systemic scale.
He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions.
For any real or complex value of z, the exponential function is defined by the equation
This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm.
The exponential function extends to an entire function on the complex plane.
In complex dynamic systems such as capital markets, risk is an exponential function of system scale.
The periodicity of an exponential function of a complex variable, along with the function's other properties, accounts for the exceptionally important role the function plays in the study of all periodic processes, particularly oscillations and wave propagation.
Периодичность Показательная функция комплексного переменного наряду с другими её свойствами является причиной, по которой эта функция играет исключительно важную роль при изучении всяких периодических процессов, в частности колебаний и распространения волн.
A more detailed account of the role of the logarithm and exponential functions in the development of complex numbers may be found in Cajori (1913).
In the real and complex numbers, addition and multiplication can be interchanged by the exponential function: ea + b = ea eb.
Сложение и умножение действительных или комплексных чисел можно взаимно заменять при помощи экспоненциальной функции: ёа + Ь = ёа ёЬ.
As you can see, since the cosine and sine functions are periodic, and the complex exponential above is made up of cosine/sine waves, then the above (actually Euler's formula) has the following property.
Как Вы видите, так как косинус и функции синуса периодические, и комплекс, показательный выше, составлен из волн косинуса/синуса, тогда у вышеупомянутого (фактически формула Эйлера) есть следующая собственность.