Complex Numbers
Let $a$ and $b$ be real numbers, the complex number is an expression in the form $a+b\mathbf{i}$ where $\mathbf{i}=\sqrt{-1}$. In the complex plane or Cartesian coordinate system, the complex number $a+b\mathbf{i}$ can represent the point with coordinate $(a,b)$.
This complex number calculator supports four binary operations: addition, substraction, multiplication, division, and the following 19 functions.
| function | description/example |
|---|---|
| $\mathrm{arg}(x)$ | This function returns the phase angle (in radians) of complex number $x$ |
| $\mathrm{abs}(x)$ | If $x=a+b\mathbf{i}$, this function returns $\sqrt{a^2+b^2}$ |
| $\mathrm{conj}(x)$ | If $x=a+b\mathbf{i}$, this function returns $a-b\mathbf{i}$ |
| $\mathrm{sqrt}(x)$ | the square root of complex number $x$ |
| $\mathrm{pow}(x,y)$ or $x\hat{\;} y$ | Calculate $x^y$, where $x$ and $y$ are complex numbers. |
| $\exp(x)$ | The base-e exponential function of complex number $x$ |
| $\log(x)$ | The natural logarithm function of complex number $x$ |
| $\sin(x)$ | The sine function of complex number $x$ |
| $\cos(x)$ | The cosine function of complex number $x$ |
| $\tan(x)$ | The tangent function of complex number $x$ |
| $\mathrm{asin}(x)$ | The inverse sine function of complex number $x$ |
| $\mathrm{acos}(x)$ | The inverse cosine function of complex number $x$ |
| $\mathrm{atan}(x)$ | The inverse tangent function of complex number $x$ |
| $\sinh(x)$ | The hyperbolic sine function of complex number $x$ |
| $\cosh(x)$ | The hyperbolic cosine function of complex number $x$ |
| $\tanh(x)$ | The hyperbolic tangent function of complex number $x$ |
| $\mathrm{asinh}(x)$ | The inverse hyperbolic sine function of complex number $x$ |
| $\mathrm{acosh}(x)$ | The inverse hyperbolic cosine function of complex number $x$ |
| $\mathrm{atanh}(x)$ | The inverse hyperbolic tangent function of complex number $x$ |
Input Data
This calculator requires two input data. The first are the data of complex numbers that will be used in the evaluation. They are entered as $n\times 2$ real matrix, where $n$ is the number of complex numbers we want to use, the first column of the matrix are the real parts and the second column the imaginary parts. The second input is a math expression of the complex numbers, where $x_i$ in the expression is the complex number in the $i^\mathrm{th}$ row of the matrix.
As an example, we want to evaluate $\exp(1+2\mathbf{i})/(3+4\mathbf{i})$. We can denote $x_1=1+2\mathbf{i}$ and $x_2=3+4\mathbf{i}$ so the math expression we want to evaluate is $\exp(x_1)/x_2$. Here is the input matrix
1 2
3 4
and the math expression
exp(x1)/x2
getting the result
0.2597 0.4776
which are the real part and imaginary part of the evaluation.
Another example, let's approximate the derivative of a function using Complex Step Differentiation. Let $F(x)=\displaystyle\frac{\mathrm{e}^x}{(\cos x)^3+(\sin x)^3}$, we want to find $F'(\pi/4)$ with complex step $h=10^{-16}$. Here is the input matrix
pi/4 1.0e-16
and the math expression
exp(x1)/( cos(x1)^3 + sin(x1)^3 )We get the output $3.101766393836052+ 3.101766393836053\times 10^{-16}\mathbf{i}$. Dividing the imaginary part with $h$ gives $3.101766393836053$, an approximated value of $F'(\pi/4)$.