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Functions of a Complex Variable Calculator

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Example
Created on 2024-06-20Asked by Madison Baker (Solvelet student)
Evaluate the complex function f(z)=z2 f(z) = z^2 at z=1+i z = 1 + i .

Solution

To evaluate the complex function f(z)=z2 f(z) = z^2 at z=1+i z = 1 + i : f(1+i)=(1+i)2 f(1 + i) = (1 + i)^2 =12+2⋅1⋅i+i2 = 1^2 + 2 \cdot 1 \cdot i + i^2 =1+2i+(−1) = 1 + 2i + (-1) =2i = 2i Solved on Solvelet with Basic AI Model
Some of the related questions asked by Theodore Taylor on Solvelet
1. Determine whether the function f(z)=z2 f(z) = z^2 is analytic in the complex plane.2. Compute the contour integral ∮(z3+2z) dz \oint (z^3 + 2z) \, dz along the unit circle.
DefinitionDefinition: In the field of complex analysis, functions of a complex variable are functions that map complex numbers to complex numbers. They are expressed using complex variables z=x+iy, with x and y being real numbers, and i the imaginary unit. These functions include holomorphic and non-holomorphic (Analytic and non-analytic) functions of a complex variable in complex analysis. They are useful in the study of complex analysis, mathematical physics, and engineering for solving problems related to the modeling of physical phenomena, solving differential equations, and analyzing complex systems. The canonical example of an entire function takes a complex number z to its exponential values, f(z)=ez, where e is the natural logarithm.
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