Complex Analytic Function Calculator

Calculator Inputs

Choose a supported analytic family, enter the complex point z = x + iy, and optionally define polynomial coefficients or integer powers.

Function Type

Real Part of z

Imaginary Part of z

Integer Power n

Contour Radius for Mapping Plot

Plot Sample Count

Polynomial Coefficients

a0 + a1z + a2z² + a3z³ + a4z⁴

a0 Real

a0 Imaginary

a1 Real

a1 Imaginary

a2 Real

a2 Imaginary

a3 Real

a3 Imaginary

a4 Real

a4 Imaginary

Example Data Table

These sample rows illustrate typical outputs from common analytic functions at selected complex points.

Function	Point z	f(z)	f′(z)	\|f(z)\|	arg(f(z))
exp(z)	1 + 1i	1.468694 + 2.287355i	1.468694 + 2.287355i	2.718282	1.000000
sin(z)	0.5 + 1i	0.739792 + 1.031336i	1.354181 - 0.563421i	1.269760	0.948876
z²	2 - 1i	3 - 4i	4 - 2i	5.000000	-0.927295
1 / z	2 + 3i	0.153846 - 0.230769i	-0.029586 + 0.071006i	0.277350	-0.982794

Formula Used

Complex point: z = x + iy

Analytic test: If f(z) = u(x, y) + iv(x, y), then the Cauchy-Riemann equations are u_x = v_y and u_y = -v_x.

Supported functions and derivatives:
exp(z) → derivative exp(z)
sin(z) → derivative cos(z)
cos(z) → derivative -sin(z)
z^n → derivative n·z^(n-1)
1 / z → derivative -1 / z²
Log(z) → derivative 1 / z on its principal branch domain
Polynomial a0 + a1z + a2z² + a3z³ + a4z⁴ → derivative a1 + 2a2z + 3a3z² + 4a4z³

Magnitude and argument:
|f(z)| = sqrt(Re(f)^2 + Im(f)^2)
arg(f(z)) = atan2(Im(f), Re(f))

Conformal behavior:
A function is locally conformal at a point when it is analytic there and f′(z) ≠ 0.

How to Use This Calculator

Select the function family you want to analyze.
Enter the real and imaginary parts of the complex input point.
For z^n, supply an integer power. For a polynomial, enter the real and imaginary coefficient parts.
Choose a contour radius and sampling count for the mapping graph.
Press Calculate Function to view f(z), f′(z), analytic checks, and the mapped contour.
Use the CSV and PDF buttons after calculation to export the computed summary.

FAQs

1. What does this calculator evaluate?

It evaluates selected complex analytic function families at a chosen point. It also computes the derivative, modulus, argument, Cauchy-Riemann residuals, and a mapped contour plot.

2. Why is the derivative important in complex analysis?

The complex derivative captures local stretching and rotation. When it exists and is nonzero, the mapping is locally conformal and preserves angles.

3. What is the Cauchy-Riemann check used for?

It numerically tests whether the real and imaginary parts satisfy the required relations for analyticity. Small residuals suggest the function behaves analytically at the selected point.

4. Why can Log(z) fail the analytic check?

The principal logarithm has a branch cut on the non-positive real axis. Values may still be displayed, but analyticity is not valid on that cut or at zero.

5. What does the mapped contour plot show?

It shows how a small circle around the chosen point transforms under the selected function. This helps visualize stretching, rotation, folding, and local distortion behavior.

6. Can I analyze custom polynomials?

Yes. Enter real and imaginary parts for coefficients a0 through a4. The calculator then evaluates the polynomial, its derivative, and the local mapping behavior.

7. When is a function locally conformal?

A function is locally conformal when it is analytic at the point and its derivative is not zero. The calculator reports this condition in the result summary.

8. Why would the calculator return undefined?

Undefined results usually occur at singular points, such as z = 0 for 1/z or Log(z). Negative powers can also fail there because division by zero appears.