Assume that Φ is harmonic in domain D and for x0 ∈ D, B(x0, r) ⊆ – McqOptions

MCQOPTIONS

1.	Assume that Φ is harmonic in domain D and for x0 ∈ D, B(x0, r) ⊆ D, then the average value of Φ over the boundary of B(x0, r) equals
A.	\({\rm{\Phi }}\left( {{x_0}} \right)\)
B.	\(r{\rm{\Phi }}\left( {{x_0}} \right)\)
C.	\(\frac{1}{{\pi r}}{\rm{\Phi }}\left( x \right)\)
D.	\(\frac{4}{3}\pi {r^3}{\rm{\Phi }}\left( x \right)\)
Answer» B. \(r{\rm{\Phi }}\left( {{x_0}} \right)\)

Discussion

No Comment Found

Related MCQs

Polar form of the Cauchy-Riemann equations is
If f(z) = (x2 + ay2) + i bxy is a complex analytic function of z = x + iy, where \(i = \sqrt { - 1} \), then
Assuming \(i = \sqrt { - 1} \) and t is a real number, \(\mathop \smallint \limits_0^{\frac{\pi}{3}} {e^{it}}dt\) is:
\(1\;+\;\frac{x^2}{2!}\;+\;\frac{x^4}{4!}\;+\;\frac{x^6}{6!}\;+\;.....\) stands for.
If f(z) is an analytic function whose real part is constant then f(z) is
Let (-1 - j), (3 - j), (3 + j) and (-1 + j) be the vertices of a rectangle C in the complex plane. Assuming that C is traversed in counter-clockwise direction, the value of the countour integral \(\displaystyle\oint_C \dfrac{dz}{z^2 (z-4)}\) is
If then the value of xx is
In the neighbourhood of z = 1, the function f(z) has a power series expansion of the form
An analytic function of a complex variable z = x + iy is expressed as f (z) = u(x,y) + iv (x, y) where i = √-1 . If u = xy, the expression for v should be
If C is a circle of radius r with center z0, in the complex z-plane and if n is a non-zero integer, then \(\mathop \oint \nolimits_C \frac{{dz}}{{{{\left( {z - {z_0}} \right)}^{n + 1}}}}\) equals