Man best?tige durch direkte Rechnung, daB die Real- und Imaginärteile der folgenden Funktionen f

Man best?tige durch direkte Rechnung, daB die Real- und...

Key Concepts

Analytic Functions

An analytic function, also known as a holomorphic function, is one that is complex differentiable in a neighborhood of every point within its domain. The property of analyticity implies that the function can be locally expressed as a convergent power series, and it inherently satisfies the Cauchy-Riemann equations.

Direct Verification Method

This method involves explicitly calculating the partial derivatives of the function's real and imaginary parts and verifying that they satisfy the Cauchy-Riemann equations. Direct verification is a straightforward computational procedure to confirm the analyticity of a function.

Real and Imaginary Parts

Any complex function can be written as the sum of its real and imaginary components. Analyzing these parts individually by computing their partial derivatives is key to applying the Cauchy-Riemann conditions, which ultimately determines whether the function is analytic.

Cauchy-Riemann Equations

These are a pair of partial differential equations that provide a necessary and sufficient condition for a function of a complex variable to be differentiable (or analytic). They relate the partial derivatives of the real and imaginary parts of the function and serve as a litmus test for determining if a complex function is holomorphic.

Transcript

00:01 Function is given f x comma y comma z equal to x square time y plus x time y plus x time y square plus y times z cube plus x time y time e to the power 2 times z first of all, we have to find the first partial derivative of f with respect to x at a point x comma y, comma z equal to del by del x of x of x square times y plus x times y square plus y times y cube plus x time y time e to the power two times z taking y z as a constant so now we differentiate it with respect to x so here we get a x sub x at a point x comma y comma z equal to in the first term y is a constant so we put here and we differentiate x square with respect to x so here we get two time x plus y square plus zero plus y time a to the power two times z so you can write like that two time xy plus y square plus y time e to the power two times z now we have to find the second order partial derivative so here f x x equal to del by del z of first partial derivative of f with respect to x so here f x equal to del by del z now we put the value of f x which is two time x y plus y square plus y time e to the power to z and here x and y as a constant so we differentiate it with respect to z so here we get first second term is zero because it is a constant because we know that the derivative of constant term is value is zero so now we put here zero plus zero now here why is a constant so we put here and the derivative of e to the the power 2 times z so value is 2 time e to the power 2 times z so here f x x equal to 2 time y time e to the power 2 z it is our answer now we find first partial derivative of f with respect y at a point x comma y x equal to del y del y del y of x square time y plus x time y plus y time set cube plus x time set cube plus x time y time e to the power to z now here taking x and z as a constant so now we differentiate it so here we get in the first term x square is constant so you put here now we have to differentiate y with respect to y so here we get one so now in the second term here we get x is a constant and we have to…

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