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Complex Functions Calculator

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Example
Created on 2024-06-20Asked by Daniel Smith (Solvelet student)
Determine whether the function f(z)=z2+zˉ f(z) = z^2 + \bar{z} is analytic.

Solution

Step 1: Identify the function f(z)=z2+zˉ f(z) = z^2 + \bar{z} . Step 2: Use the Cauchy-Riemann equations to determine analyticity. Let z=x+iy z = x + iy , then f(z)=(x+iy)2+(xiy)=x2y2+2ix+xiy f(z) = (x + iy)^2 + (x - iy) = x^2 - y^2 + 2ix + x - iy . Step 3: Separate into real and imaginary parts: u(x,y)=x2y2+x,v(x,y)=2xyy. u(x,y) = x^2 - y^2 + x, \quad v(x,y) = 2xy - y. Step 4: Compute the partial derivatives: ux=2x+1,uy=2y,vx=2y,vy=2x1. u_x = 2x + 1, \quad u_y = -2y, \quad v_x = 2y, \quad v_y = 2x - 1. Step 5: Check the Cauchy-Riemann equations: ux=vyanduy=vx. u_x = v_y \quad \text{and} \quad u_y = -v_x. Step 6: Verify the equations: 2x+12x1and2y2y. 2x + 1 \neq 2x - 1 \quad \text{and} \quad -2y \neq -2y. Step 7: Conclusion. The function f(z)=z2+zˉ f(z) = z^2 + \bar{z} is not analytic because it does not satisfy the Cauchy-Riemann equations. Solved on Solvelet with Basic AI Model
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DefinitionComplex functions, unlike real functions, have the domain of such functions as complex numbers. If we begin from the definition, such a presentation can be expressed in the way of f z = u x, y + iv x, y, where f and v denote Bert the actual functions of two real variables x and y. There is no doubt such a classification as complex simple, while both and are real functions. Complex simple elementary functions, i.e., the class of elementary functions, are characterized by either being rational functions or else a polynomial dominated approach. Either way takes two distinct kinds, one being an exponential function and the other is referred to as the trigonometric function. Moreover, it is interesting to learn that the cosine function and the sine function apply to the Hewitt the classes for several individuality logarithmic particular-cosine features and the Mod of the arrangement of several individual logarithmic particular-sine elements.
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