Complex Antiderivative Calculator

Calculator Inputs

Supported patterns include constants, linear powers such as (2+i)*z^3 or (z+1)^-1, and linear functions inside exp(), sin(), cos(), sinh(), and cosh().

Complex expression f(z)

Example: exp((1+i)*z) + (2-i)*z^2 + 1/(z+1)

Variable

Single variable only, usually z.

Integration constant C

Use 0, 2, -i, or 1+3i.

Lower real bound

Used for the definite line integral check.

Upper real bound

End point is x + iy on the same slice.

Imaginary slice y

Plots and validation use z = x + iy.

Plot start x

Plot end x

Plot samples

Odd values work best for smoother checks.

Plotly Graph

The graph traces the real part, imaginary part, and magnitude of the antiderivative on the horizontal slice z = x + iy.

Example Data Table

Input f(z)	Suggested antiderivative F(z)	Rule used	Comment
(2-i)*z^2	((2-i)/3)z^3 + C	Power rule	Apply n + 1 in the exponent and divide by 3.
exp((1+i)z)	exp((1+i)z)/(1+i) + C	Linear exponential rule	Divide by the inner derivative coefficient.
1/(z+1)	Log(z+1) + C	Log rule	Uses the principal logarithm branch for complex values.
cos((2-i)z+3)	sin((2-i)z+3)/(2-i) + C	Chain-rule reversal	Differentiate the proposed result to verify the input.

Formula Used

∫ c(a z + b)ⁿ dz = c(a z + b)ⁿ⁺¹ / [a(n+1)], for n ≠ -1

∫ c/(a z + b) dz = (c/a) Log(a z + b)

∫ c·exp(a z + b) dz = (c/a) exp(a z + b)

∫ c·sin(a z + b) dz = -(c/a) cos(a z + b)

∫ c·cos(a z + b) dz = (c/a) sin(a z + b)

Complex plots evaluate F(z) on z = x + iy, while definite validation compares F(z₂) − F(z₁) with Simpson integration along that same horizontal path.

How to Use This Calculator

Enter the complex expression using the supported syntax. Put complex coefficients in parentheses when needed.
Choose the variable, then set the constant C if you want a specific antiderivative family member.
Provide lower and upper real bounds plus an imaginary slice to test a definite change on z = x + iy.
Set the plot start, plot end, and number of samples for the graph.
Press Submit. The result appears above the form and below the header.
Review the symbolic answer, derivative check, and numerical validation. Then export the table as CSV or PDF.

FAQs

1) What expressions does this calculator support?

It supports sums of constants, powers of linear terms, reciprocal linear terms, and linear arguments inside exp, sin, cos, sinh, and cosh. Expressions outside those families should be simplified or entered in supported pieces.

2) Does it handle complex coefficients?

Yes. You can enter values like 2-i, 3+4i, or -i. Wrapping complex coefficients in parentheses makes parsing clearer, especially before powers or functions.

3) Why does the logarithm answer matter in complex analysis?

Complex logarithms are multi-valued. This page uses the principal branch for numerical evaluation, so branch cuts can affect displayed values even when symbolic differentiation still returns the original reciprocal term.

4) What does the derivative check do?

It numerically differentiates the computed antiderivative at one sample point and compares that estimate with the original integrand. A small gap suggests the symbolic result and numerical evaluation are aligned.

5) What is the definite value section showing?

It calculates F(z₂) − F(z₁) along the horizontal path z = x + iy and compares that result with Simpson integration of f(z) over the same line segment.

6) Why plot real and imaginary parts separately?

A complex antiderivative has two visible coordinate components. Plotting both helps you inspect oscillation, growth, and branch behavior more clearly than looking at magnitude alone.

7) Can I use a different variable than z?

Yes. Enter one letter as the variable name. The symbolic rules stay the same, but all supported expressions must consistently use that same variable throughout the input.

8) What should I do if a term is rejected?

Rewrite the expression into supported parts, add parentheses around complex coefficients, or split a large expression into separate terms. That usually resolves syntax problems and clarifies the intended structure.