By Michael J. Thompson

This booklet offers an creation for graduate scholars and complicated undergraduate scholars to the sector of astrophysical fluid dynamics. even supposing occasionally overlooked, fluid dynamical tactics play a critical position in almost all components of astrophysics. No prior wisdom of fluid dynamics is thought. After setting up the fundamental equations of fluid dynamics and the physics correct to an astrophysical program, various issues within the box are addressed. there's additionally a bankruptcy introducing the reader to numerical equipment. Appendices record precious actual constants and astronomical amounts, and supply convenient reference fabric on Cartesian tensors, vector calculus in polar coordinates, self-adjoint eigenvalue difficulties and JWKB concept.

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An ∈ C such that P (z) a2 an a1 + + ... + . = Q(z) z − z1 z − z2 z − zn a) We shall ﬁrst of all show that the expression above is possible by multiplying it by Q(z) and then determining a1 , a2 , . . , an so that the resulting equation between polynomials of degree less than n holds when z = z1 , z2 , . . , zn . ] b) Show that for every k = 1, . . , n, we have ak = lim (z − zk ) z→zk P (zk ) P (z) = . Q(z) Q (zk ) [Hint: Note that Q(zk ) = 0 for every k = 1, . . ] Chapter 3 : Complex Diﬀerentiation 3–15 8.

Suppose that u is a real valued harmonic function in a domain D. Write (21) g(z) = ∂u ∂u −i . ∂x ∂y Then the Cauchy-Riemann equations for g are ∂ ∂x ∂u ∂x =− ∂ ∂y ∂u ∂y and ∂ ∂y ∂u ∂x = ∂ ∂x ∂u ∂y , which clearly hold. It follows that g is analytic in D. Suppose now that u is the real part of an analytic function f in D. Then f (z) agrees with the right hand side of (21) in view of (3) and (4). Hence f = g Chapter 3 : Complex Diﬀerentiation 3–13 in D. The question here, of course, is to ﬁnd this function f .

Indeed, if the functions f : D → C and g : D → C are both diﬀerentiable at z0 ∈ D, then both f + g and f g are diﬀerentiable at z0 , and (f + g) (z0 ) = f (z0 ) + g (z0 ) and (f g) (z0 ) = f (z0 )g (z0 ) + f (z0 )g(z0 ). 3–2 W W L Chen : Introduction to Complex Analysis If the extra condition g (z0 ) = 0 holds, then f /g is diﬀerentiable at z0 , and f g (z0 ) = g(z0 )f (z0 ) − f (z0 )g (z0 ) . g 2 (z0 ) One can also establish the Chain rule for diﬀerentiation as in real analysis. More precisely, suppose that the function f is diﬀerentiable at z0 and the function g is diﬀerentiable at w0 = f (z0 ).

### An Introduction to Astrophysical Fluid Dynamics by Michael J. Thompson

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