Complex Numbers |
MORE ON COMPLEX NUMBERS |
Note: Some textbooks use the letter j to represent the imaginary part of a complex number. I have used the more universal i throughout. |
A complex number, z, is of the form: |
![](../../eqn/0337.gif) |
or, using polar coordinates: |
![](../../eqn/0338.gif) |
where x and y are real numbers and: |
![](../../eqn/0339.gif) |
![](../../eqn/0340.gif) |
The modulus of a complex number is: |
![](../../eqn/0341.gif) |
The argument of a complex number is: |
![](../../eqn/0342.gif) |
The conjugate of a complex number is: |
![](../../eqn/0343.gif) |
In the following lines: |
![](../../eqn/0344.gif) |
![](../../eqn/0345.gif) |
![](../../eqn/0346.gif) |
![](../../eqn/0347.gif) |
![](../../eqn/0348.gif) |
![](../../eqn/0349.gif) |
For z = x + iy and where y is measured in radians: |
![](../../eqn/0350.gif) |
When is measured in radians: |
The exponential form of a complex number: |
![](../../eqn/0351.gif) |
![](../../eqn/0357.gif) |
![](../../eqn/0358.gif) |
![](../../eqn/0359.gif) |
![](../../eqn/0360.gif) |
![](../../eqn/0356.gif) |
![](../../eqn/0352.gif) |
![](../../eqn/0355.gif) |
Complex Trigonometric Functions: |
![](../../eqn/0361.gif) |
![](../../eqn/0362.gif) |
![](../../eqn/0363.gif) |
![](../../eqn/0364.gif) |
![](../../eqn/0365.gif) |
![](../../eqn/0366.gif) |
Complex Trigonometric Function Properties: |
![](../../eqn/0367.gif) |
![](../../eqn/0368.gif) |
![](../../eqn/0369.gif) |
![](../../eqn/0370.gif) |
![](../../eqn/0371.gif) |
![](../../eqn/0372.gif) |
![](../../eqn/0373.gif) |
![](../../eqn/0374.gif) |
![](../../eqn/0375.gif) |
Complex Hyperbolic Functions: |
![](../../eqn/0376.gif) |
![](../../eqn/0377.gif) |
![](../../eqn/0378.gif) |
![](../../eqn/0379.gif) |
![](../../eqn/0380.gif) |
![](../../eqn/0381.gif) |
Complex Hyperbolic Properties: |
![](../../eqn/0382.gif) |
![](../../eqn/0383.gif) |
![](../../eqn/0384.gif) |
![](../../eqn/0385.gif) |
![](../../eqn/0386.gif) |
![](../../eqn/0387.gif) |
![](../../eqn/0388.gif) |
![](../../eqn/0389.gif) |
![](../../eqn/0390.gif) |
Complex Trigonometric and Hyperbolic Function Relations: |
![](../../eqn/0391.gif) |
![](../../eqn/0392.gif) |
![](../../eqn/0393.gif) |
![](../../eqn/0394.gif) |
![](../../eqn/0395.gif) |
![](../../eqn/0396.gif) |
Complex Inverse Trigonometric Functions: |
![](../../eqn/0397.gif) |
![](../../eqn/0398.gif) |
![](../../eqn/0399.gif) |
![](../../eqn/0400.gif) |
![](../../eqn/0401.gif) |
![](../../eqn/0402.gif) |
Complex Inverse Hyperbolic Functions: |
![](../../eqn/0403.gif) |
![](../../eqn/0404.gif) |
![](../../eqn/0405.gif) |
![](../../eqn/0406.gif) |
![](../../eqn/0407.gif) |
![](../../eqn/0408.gif) |