Define Harmonic (or) Potential function.
Answers
Answer:
Step-by-step explanation:
Harmonic function, mathematical function of two variables having the property that its value at any point is equal to the average of its values along any circle around that point, provided the function is defined within the circle.
Please mark me as brainliest....
Answer:
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R where U is an open subset of Rn that satisfies Laplace's equation, that is,
{\displaystyle {\frac {\partial ^{2}f}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}+\cdots +{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}=0} \frac{\partial^2f}{\partial x_1^2} + \frac{\partial^2f}{\partial x_2^2} + \cdots + \frac{\partial^2f}{\partial x_n^2} = 0
everywhere on U. This is usually written as
{\displaystyle \nabla ^{2}f=0} \nabla^2 f = 0
or
{\displaystyle \textstyle \Delta f=0} \textstyle \Delta f = 0
Step-by-step explanation:
Examples of harmonic functions of two variables are:
The real and imaginary parts of any holomorphic function
The function {\displaystyle \,\!f(x,y)=e^{x}\sin y} \,\! f(x,y)=e^{x} \sin y; this is a special case of the example above, as {\displaystyle f(x,y)=\operatorname {Im} (e^{x+iy})} f(x,y)=\operatorname {Im}(e^{{x+iy}}), and {\displaystyle e^{x+iy}} e^{x+iy} is a holomorphic function.
The function {\displaystyle \,\!f(x,y)=\ln(x^{2}+y^{2})} {\displaystyle \,\!f(x,y)=\ln(x^{2}+y^{2})} defined on {\displaystyle \mathbb {R} ^{2}\setminus \lbrace 0\rbrace } {\mathbb {R}}^{2}\setminus \lbrace 0\rbrace . This can describe the electric potential due to a line charge or the gravity potential due to a long cylindrical mass.
hope this help you p
plz mark me as brainliest