Complex Exponential Calculator

Evaluate e^(a+bi) using clear steps and flexible units. See plots, identities, and fast downloadable summaries. Perfect for students, engineers, analysts, and curious problem solvers.

Calculator Inputs

Real part a

Imaginary coefficient b

Input angle unit for b

Preferred angle display

Decimal precision

Plot resolution points

Reset

Example Data Table

These examples show how different values of a and b change the final complex exponential.

Real part a	Imaginary part b	Unit	Rectangular form of e^(a+bi)	Magnitude	Angle
0	π/2 rad	rad	0.000000 + 1.000000i	1.000000	89.999981°
1	45°	deg	1.922116 + 1.922116i	2.718282	45.000000°
-0.5	π rad	rad	-0.606531 - 0.000000i	0.606531	-179.999980°
2	-60°	deg	3.694528 - 6.399110i	7.389056	-60.000000°

Formula Used

Main identity: e^(a+bi) = e^a(cos b + i sin b)

This is Euler’s formula applied to a complex exponent. The real part a controls scaling, while the imaginary part b controls rotation.

Rectangular output: Re = e^a cos b and Im = e^a sin b

The calculator uses these two equations to build the final value in standard complex form.

Magnitude: |e^(a+bi)| = e^a

Because cos²b + sin²b = 1, the magnitude depends only on the real part.

Argument: arg(e^(a+bi)) = b modulo 2π

The graph wraps angle values to the principal interval when reporting the principal argument.

How to Use This Calculator

Step 1

Enter the real part a and the imaginary coefficient b.

Step 2

Choose whether b is given in radians or degrees.

Step 3

Set the decimal precision and plot resolution for cleaner results and smoother graphing.

Step 4

Press the calculate button. The result appears above the form, under the header.

Step 5

Review rectangular, polar, and Euler forms, then export the summary as CSV or PDF.

Frequently Asked Questions

1) What does e^(a+bi) represent?

It is the exponential function evaluated at a complex number. The value combines growth from the real part and rotation from the imaginary part into one complex result.

2) Why does the magnitude depend only on a?

Because the trigonometric part has unit magnitude. Euler’s identity keeps the circular component on a radius of one, so only the factor e^a changes the final magnitude.

3) Can I enter degrees instead of radians?

Yes. Select degrees from the unit menu. The calculator converts the angle internally before applying the exponential identity and then reports the principal argument clearly.

4) What does the graph show?

The graph shows the final point on the complex plane, the circular path traced by the angle, and the full circle whose radius equals e^a.

5) Why are rectangular and polar forms both useful?

Rectangular form is best for addition and component reading. Polar form is better for magnitude, angle, scaling, and geometric interpretation on the complex plane.

6) Why is the derivative listed as the same value?

The complex exponential has the same derivative as its value. So for any input z, the derivative of e^z is simply e^z again.

7) When does the result lie on the unit circle?

It lies on the unit circle when a equals zero. In that case, e^a becomes one, leaving only the pure rotational part cos b + i sin b.

8) Why export CSV or PDF?

CSV is helpful for spreadsheets and repeated comparisons. PDF is useful for sharing a neat report, saving class work, or attaching results to documentation.