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1
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2.9k
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1
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Find the bilienar transformation which maps the point z = -1, 1, $\infty$ onto the points $w = -i, -1, i$
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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1.9k
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Under the transformation $w + zi = z + \frac{1}{z}$. Show that the map of the circle |z| = z is an ellipse in $w$-plane.
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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Find the bilinear transformation under which 1,i, -1 from z plane are mapped onto 0,1,$\infty$ of w plane. Further show the following.
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
3
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68k
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Find the bilinear transformation which maps the points z = 1, I, -1 onto w = 0,1,$\infty$
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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1
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8.1k
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Show that the function $u = sinx coshy + 2cosx sinh y + x^2 - y^2 + 4xy$ satisfies Laplace's equation and find its corresponding analytic function.
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
3
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13k
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If $v = e^x siny$, Prove that v is a harmonic function. Also find the corresponding harmonic conjugate function and analytic function.
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
2
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1
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20k
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Determine the constants a,b,c,d if $f(z) = x^2 + 2axy + by^2 + i(dx^2 + 2cxy + y^2)$ is analytic.
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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6.1k
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Prove that $f(z) = x^2 - y^2 + 2ixy$ is analytic and find $f'(z)$.
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
1
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25k
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Define bilinear transformation and prove that in a general a bilinear transformation maps into a circle.
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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17k
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Find the analytic function $f(z) = u + iv$ where $u = r^2 cos2\theta - rcos\theta + 2$
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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4.8k
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Find the orthogonal trajectory of the curves $3x^2 y + 2x^2 - y^3 - zy^2 = \alpha$ where $\alpha$ is constant.
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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Show that under the transformation w = (5 - 4z)/(4z - 2) the circle |z| = 1 in the z plane is transformed into a circle of unity in the W plane.
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
1
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1
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4.0k
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1
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Evaluate $\int J_{3}(x)dx$ and express the result in terms of $J_{0}$ and $J_{1}$
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
3
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1
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23k
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1
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Show that the function $v = e^x (xsiny + ycosy)$ satisfies Laplace equation. Find its corresponding analytic function & its harmonic conjugate.
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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1.5k
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Show that $J_{\frac{5}{2}} = \sqrt{\frac{2}{\pi x}}\bigg[\frac{3 - x^2}{x^2}sinx - \frac{3}{2}cosx \bigg]$
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
3
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12k
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Find the imaginary part whose real part is $u = x^3 - 3xy^2 + 3x^2 - 3y^2 + 1$
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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1.9k
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Prove that $\int J_{3}(x)dx = \frac{-2J_{1}(x)}{x} - J_{2}(x)$
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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24k
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Prove that real and imaginary parts of an analytic function f(z) = u + iv are harmonic function.
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
3
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1
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20k
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1
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Using Gauss divergence theorem evaluate $\int\limits_{C}\bar{F} \cdot \overline{ds}$ where $F = 2x^2yi - y^2j + 4xz^2k$ and S is the region bounded by $y^2 + z^2 = 9$ and x = 2 in the first octant.
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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1
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4.6k
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1
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Verify green's theorem in the plane for $\oint\limits_{s} (x^2 - y)dx + (2y^2 + x)dy$ around the boundary of region defined by $y = x^2$ and y = 4
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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1
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2.5k
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1
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Evaluate by stoke's theorem $\int\limits_{C}xydx + xy^2 dy$ where C is the square in the xy plane with vertices (1,0), (0,1), (-1,0) and (0,1)
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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1.7k
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Using Green's theorem evaluate $\int\limits_{c}(2x^2 - y^2) + (x^2 - y^2)dy$ where c is the boundary of the surface enclosed by the lines x = 0, y = 0, x = 2, y = 2
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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14k
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1
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Define Solenoidal vector. Hence prove that $\bar{F} = \frac{\bar{a} \times \bar{r}}{r^n}$ is a solenoidal vector.
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
3
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27k
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1
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A vector field is given by $\bar{F} = (x^2 + xy^2)i + (y^2 + x^2 y)j$. Show that $\bar{F}$ is irrotational and find its scalar potential such that $\bar{F} = \bar{V}\phi$
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
1
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7.1k
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1
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Find the directional derivative of $\phi = x^4 + y^4 + z^4$ at A(1,-2,1) in the direction of AB where B is (2,6,-1). Also find the maximum directional derivative of $\phi$ at (1,-2,1)
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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