1. THEORY: Analytic function (Definition ), derivative of a function of complex variable (Definition, calculation ),...

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1. THEORY: Analytic function (Definition ), derivative of a function of complex variable (Definition, calculation ), necessary condition of existence of derivative; Task: Check out if the Cauchy-Riemann equations are satisfied for the given function. Calculate f(2) if it exists. f(2) = cos z +1. 2. Evaluate the complex line integral [(2 Rez+z)dz, if L is the line y= x', connecting the point z, = 2+4i with the point z, =0. 3. Find a circulation of the vector field a =-24yi +87+6zk along the triangle contour ABC: A(8,0,0), B(0,2,0), C(0,0,4). 4. Find a mass of a material line L, if L is a function's y= tan x graph in the y cos x Vi+ cosx interval 0sx S , and the line's density function is p(x, y) = 4 5. Find a flux of the vector field a = xi -xj+2k through the lateral surface of the cone S: x +y = z, restricted by the plane S1: z=5. 6. Using the Laplace transform, solve the differential equation, and check the correctness of the obtained solution. y"+25y = e", y(0) = 0, y'(0)=1. 1. THEORY: Analytic function (Definition ), derivative of a function of complex variable (Definition, calculation ), necessary condition of existence of derivative; Task: Check out if the Cauchy-Riemann equations are satisfied for the given function. Calculate f(2) if it exists. f(2) = cos z +1. 2. Evaluate the complex line integral [(2 Rez+z)dz, if L is the line y= x', connecting the point z, = 2+4i with the point z, =0. 3. Find a circulation of the vector field a =-24yi +87+6zk along the triangle contour ABC: A(8,0,0), B(0,2,0), C(0,0,4). 4. Find a mass of a material line L, if L is a function's y= tan x graph in the y cos x Vi+ cosx interval 0sx S , and the line's density function is p(x, y) = 4 5. Find a flux of the vector field a = xi -xj+2k through the lateral surface of the cone S: x +y = z, restricted by the plane S1: z=5. 6. Using the Laplace transform, solve the differential equation, and check the correctness of the obtained solution. y"+25y = e", y(0) = 0, y'(0)=1.