課程名稱︰應用數學三
課程性質︰物理系必帶
課程教師︰蔡爾成
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰2019年4月18日
考試時限（分鐘）：150分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Applied Mathematics III 4-18-2019
1.[10%] Show that ▽(A‧B)=Ax(▽xB)+Bx(▽xA)+(A‧▽)B+(B‧▽)A
2.[10%] Show that the real and imaginary parts of any twice-differentiable fu-
ction fo the form f(z) satisfy Laplace's equation and the Cauchy-Riemann eq-
uations are satisfied if f(z) is an analytic function of z.
3.[10%] Show that for the function
w =arcsin(z)+ilog(z)
the point z∞ is a branch point only on those branches for which w does not
remain finite as z→z∞.
4.[10%] Prove that a function which is analytic at all finite points and at
z∞ must be a constant.
5.[10%] Prove that any polynomial of degree N, f(z)=a0+a1z+...+aNz^N, has at
least one zero unless it is a constant.
6.(a)[5%] Show that
∮ f(z)dz=∮ f(1/t)t^(-2)dt
|z|=R |t|=1/R
By letting R→∞, deduce that ∮_{C∞}f(z)dz is given by 2πi times the
residue of 1/t^2f(1/t) at t=0.
(b)[5%] Use the above result to evaluate the integral
a^2-z^2 dz
∮──── ──
c a^2+z^2 z
where C is any positive contour enclosing the points z=0,±ia, and check
the result by calculating the residues at thos poles.
7.[10%] Using residue calculus to show that
∞ x^2 dx π
∫ ────────────── = ─────
-∞ (x^2+1)(x^2-2xcos(ω)+1) 2|sin(ω)|
if is ω real and sin(ω)≠0.
8.[10%] Show that
∞ e^(px) π
∫ ──── dx = ────
-∞ 1+e^x sin(pπ)
provided that 0<Re(p)<1.
9.[10%] Show that A vector is uniquely specified by giving its divergence and
its divergence and its curl within a simply connected region (without holes)
R and its normal component over the boundary. i.e., show that if ▽‧V1=
▽‧V2, ▽xV1=▽xV2 in R and n‧V1=n‧V2 on ∂R, then V1=V2.
10.[10%] Suppose that, on a circular arc C_R with radius R and center at the
origin, f(z)→0 uniformly as R→∞. If C_R is in the first and/or fourth
quadrants, show that
lim ∫ e^(-mz)f(z)dz = 0. (m>0)
R→∞ C_R