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- https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/02%3A_Convexity_and_Pseudoconvexity/2.04%3A_Harmonic_Subharmonic_and_Plurisubharmonic_FunctionsWe remark that when n=1 in the definition of a subharmonic function, it is the same as the standard definition of a convex function of one real variable, where affine linear functions play the rol...We remark that when n=1 in the definition of a subharmonic function, it is the same as the standard definition of a convex function of one real variable, where affine linear functions play the role of harmonic functions: A function of one real variable is convex if for every interval it is less than the affine linear function with the same end points.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/02%3A_Second_Order_Partial_Differential_Equations/2.05%3A_Laplaces_Equation_in_2DAnother generic partial differential equation is Laplace’s equation, ∇²u=0 . Laplace’s equation arises in many applications. Solutions of Laplace’s equation are called harmonic functions.
- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/06%3A_Harmonic_Functions/6.02%3A_Harmonic_FunctionsWe start by defining harmonic functions and looking at some of their properties.
- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/06%3A_Harmonic_FunctionsHarmonic functions appear regularly and play a fundamental role in math, physics and engineering. In this topic we’ll learn the definition, some key properties and their tight connection to complex an...Harmonic functions appear regularly and play a fundamental role in math, physics and engineering. In this topic we’ll learn the definition, some key properties and their tight connection to complex analysis. The key connection to 18.04 is that both the real and imaginary parts of analytic functions are harmonic. We will see that this is a simple consequence of the Cauchy-Riemann equations. In the next topic we will look at some applications to hydrodynamics.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Differential_Equations_for_Engineers_(Lebl)/4%3A_Fourier_series_and_PDEs/4.09%3A_Steady_state_temperature_and_the_LaplacianSuppose we have an insulated wire, a plate, or a 3-dimensional object. We apply certain fixed temperatures on the ends of the wire, the edges of the plate, or on all sides of the 3-dimensional object....Suppose we have an insulated wire, a plate, or a 3-dimensional object. We apply certain fixed temperatures on the ends of the wire, the edges of the plate, or on all sides of the 3-dimensional object. We wish to find out what is the steady state temperature distribution. That is, we wish to know what will be the temperature after long enough period of time.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/06%3A_Problems_in_Higher_Dimensions/6.05%3A_Laplaces_Equation_and_Spherical_SymmetryWe have seen that Laplace's equation, ∇²u=0, arises in electrostatics as an equation for electric potential outside a charge distribution and it occurs as the equation governing equilibrium temperatu...We have seen that Laplace's equation, ∇²u=0, arises in electrostatics as an equation for electric potential outside a charge distribution and it occurs as the equation governing equilibrium temperature distributions. As we had seen in the last chapter, Laplace’s equation generally occurs in the study of potential theory, which also includes the study of gravitational and fluid potentials.
