The contour integral is to be evaluated on a circle of radius 2a centred at the origin. It will have contributions only from the points
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Mathematical Physics
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Which of the following functions of the complex variable z is not analytic everywhere?
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Consider a vector in the coordinate system The axes are rotated anti-clockwise about the Y-axis by an angle of 60°. The vector in the rotate coordinate system is
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The value of the integral where the contour C is the unit circle: |z - 2| = 1, is
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A Iinear transformation T, defined as T\left[ {\begin{array}{*{20}{c}}
{{x_1}} \\
{{x_2}} \\
{{x_3}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
{{x_1} + {x_2}} \\
{{x_2} - {x_3}}
\end{array}} \right], transforms a vector for a three-dimensional real space to a two-dimensional real space. The transformation matrix T is
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For the complex function, which of the following statement is correct?
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Which one of the following matrices is the inverse of
the matrix \left[ {\begin{array}{*{20}{c}}
1&{ - 1} \\
0&1
\end{array}} \right]?
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Let and where is the Levi-Civita density, defined to be zero, if two 'of the indices. coincide and +1 and -1 depending on whether ijk is even or odd permutation of 1, 2, 3. Then β3 is equal to
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Given the recurrence relation for the Legendre polynomials (2n + 1) xPn(x) = (n + 1) Pn + 1(x) + Pn - 1(x), which of the following integrals has a non-zero value?
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The unit vector normal to the surface 3x2 + 4y = z at the point (1, 1, 7) is
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For the function the value of at x = y = 1 is
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The eigen values of the matrix \left[ {\begin{array}{*{20}{c}}
1&i \\
{ - i}&1
\end{array}} \right] are
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If the Fourier transform F[δ(x - a)] = exp (-i2πv a), then F-1(cos 2π av) will correspond to
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The solutions to the differential equation are a family of
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The Laplace transform of f(t) = sin πt is Therefore, the Laplace transform of t sin πt is
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A finite wave train, of an unspecified nature, propagates along the positive X-axis with a constant speed v and without any change of shape. The differential equation among the four listed below, whose solution it must be, is
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The eigen values of the matrix \left[ {\begin{array}{*{20}{c}}
2&3&0 \\
3&2&0 \\
0&0&1
\end{array}} \right] are
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The value of the integral where z is a complex variable and C is the unit circle with the origin as its centre, is
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Which of the following vectors is orthogonal to the vector where a and b (a ≠ b) are constants, and and are unit orthogonal vectors?
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The two vectors are
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