Raise Complex Number to Power Calculator

Complex Power Input

Enter the base, exponent, branch, and precision. Submit to see the result above this form.

Formula Used

Let the base be z = a + bi. Let the exponent be w = c + di.

First convert the base to polar form. Use r = √(a² + b²) and θ = atan2(b, a).

Complex logarithms use branches. The selected angle is θk = θ + 2πk.

The power rule is z^w = exp(w × Log(z)).

So the calculator uses z^w = exp((c + di)(ln(r) + iθk)).

The rectangular answer is built from magnitude × cos(angle) and magnitude × sin(angle).

How to Use This Calculator

1. Enter the real and imaginary parts of the base.
2. Enter the real and imaginary parts of the exponent.
3. Set the branch value. Use zero for the principal value.
4. Choose how many nearby branches to compare.
5. Pick angle units and decimal precision.
6. Press the calculate button to see the answer.
7. Use the download buttons for saving results.

Example Data Table

Understanding Complex Powers

A complex power can look hard at first. The base has a real part and an imaginary part. The exponent may also have both parts. This calculator changes the base into polar form first. That step makes powers easier to handle. It finds the magnitude and the angle. Then it applies the logarithm rule for complex numbers.

The important idea is that a complex angle can repeat. One angle can have many equivalent values. They differ by full turns. Because of this, complex powers can have different branch results. The branch input lets you test those possibilities. A branch value of zero gives the principal value. Other branches add or subtract full rotations.

Why Polar Form Helps

Rectangular form uses a plus bi. It is simple for addition and subtraction. Polar form uses size and direction. It is better for multiplication, powers, and roots. When a complex number is raised to a power, the magnitude and angle are transformed together. The calculator shows both forms for clarity.

For integer powers, the result usually feels familiar. The magnitude is raised to that integer. The angle is multiplied by that integer. For fractional or complex powers, the logarithm becomes more important. The imaginary part of the exponent can change the final magnitude. It can also rotate the final answer.

Practical Uses

This tool is useful in algebra, signals, circuits, waves, and control systems. Complex powers appear in transfer functions and frequency response work. They also help with exponential models that rotate. Students can use the step details to check homework. Engineers can compare branch behavior before using a value in a larger model.

The calculator also helps avoid sign errors. Complex arithmetic often creates small mistakes. A clear split between real parts, imaginary parts, magnitude, and angle reduces confusion. The precision option lets you control rounding. More digits help during technical work. Fewer digits help when preparing a simple answer.

Helpful Input Choices

Use degrees when your class writes angles in degrees. Use radians when formulas need radians. Keep branch zero for normal principal work. Try branches when a lesson asks for all values. Choose higher precision when comparing small differences. Round only after the final answer appears.

Reading the Result

The rectangular result is the final a plus bi value. The polar result gives magnitude and angle. The selected branch result uses your branch number. The principal value uses branch zero. When the exponent is an integer, branches normally lead to the same visible value. When the exponent is complex, branches can change the answer.

Always check the input signs carefully. A negative imaginary part changes the angle. A zero base needs special care. Some zero base powers are undefined. The calculator warns you when the chosen operation is not valid. Use the formula section to understand each step. Careful settings make complex power work easier for everyone.

FAQs