Mathematical Tools for Physics

1—Basic Stuff 26

1.19 Show that

Γ(n+^1 / 2 ) =

√

π

2 n

(2n−1)!!

The “double factorial” symbol mean the product of every other integer up to the given one. E.g.5!! = 15.
(b) The double factorial of an even integer can be expressed in terms of the single factorial. Do so. What about
odd integers?

1.20 Evaluate this integral. Just find the right substitution.

∫∞

0

dte−t

a

(a >0)

1.21 A triangle has sidesa,b,c, and the angle oppositecisγ. Express the area of the triangle in terms ofa,b,
andγ. Write the law of cosines for this triangle and then usesin^2 γ+ cos^2 γ= 1to express the area of a triangle
solely in terms of the lengths of its three sides. The resulting formula is not especially pretty or even clearly
symmetrical in the sides, but if you introduce the semiperimeter,s= (a+b+c)/ 2 , you can rearrange the answer
into a neat, symmetrical form. Check its validity in a couple of special cases. Ans:

√

s(s−a)(s−b)(s−c)

1.22 Express an arbitrary linear combination of the sine and cosine ofθ,Asinθ+Bcosθ, as a phase-shifted
cosine:Ccos(θ+δ). Solve forCandδin terms ofAandB, deriving an identity inθ.

1.23 Solve the two simultaneous linear equations

ax+by=e, cx+dy=f

and do it solely by elementary manipulation (+,−,×,÷), not by any special formulas. Analyze all thequalitatively
differentcases and draw graphs to describe each. In every case, how many if any solutions are there? Because of
its special importance later, look at the casee=f= 0and analyze it as if it’s a separate problem. You should be
able to discern and to classify the circumstances under which there is one solution, no solution, or many solutions.

1.24 Use parametric differentiation to evaluate the integral

∫

x^2 sinxdx

Find a table of integrals if you want to verify your work.