Finding the Right Solutions Manual for Brown and Churchill
I picked up the 9th edition of Complex Variables and Applications when I was teaching the course at a community college back in 2018, and let me tell you, the solutions situation with this one is messier than most textbooks. You've got the official solutions manual from McGraw-Hill, which is decent but expensive and sometimes skips steps on the harder problems. Then there's every PDF floating around the internet that's either scanned poorly or written by someone who clearly didn't finish the problem set themselves. The official James Ward Brown And Ruel V Churchill Complex Variables And Applications 9th Edition Solutions manual covers odd-numbered problems mostly. If you're looking for even-numbered work, you're going to need the instructor's solution manual, which isn't freely available and requires a legitimate institutional request to get through proper channels. I used to grab it through our university library's reserve system before they started cracking down on digital access.
What the Official Manual Gets Right and Wrong
The odd-numbered solutions are generally accurate for problems one through three. But here's where people run into trouble — problems five and beyond in Chapter 3, especially the ones involving contour integration with branch cuts, the manual occasionally uses a different branch choice than what the textbook problems imply. I caught this on problem set 3.7, problem 11. The textbook expects you to take the principal branch of the logarithm with a cut along the negative real axis, but the solutions manual subtly switches to a cut along the positive imaginary axis midway through. That throws off your residue calculations if you don't catch it. The worked examples in the front sections are solid though. The partial fractions decomposition walkthrough on page 84 for rational function integration is worth studying carefully because the problem set builds directly on that method. Another thing the manual does well is showing the substitution z = e^{i} for integrals over the unit circle. That technique appears in Chapter 6 and shows up again in the engineering applications sections later on.
Where the Manual Falls Apart
Chapter 8 on conformal mapping is where the solutions get sparse and hand-wavy. There are maybe six worked problems in that entire chapter, and two of them just state the final mapping function without deriving how they got there. If you're struggling with the Möbius transformation problems — and most students are — the manual won't carry you far. I had a student last semester who spent four hours on problem 13 in section 8.3 because the solution just wrote "the required mapping is w = (z - i)/(z + i)" with no explanation of how the boundary correspondence was determined. Another gap: the residue calculus section in Chapter 7 has problems where the manual assumes you already know which poles lie inside your contour. That sounds obvious until you're dealing with a problem like 7.4.15 where the contour is defined parametrically and you have to figure out which singularities of tan(z) actually fall inside it. The manual just lists the answer.
Get the Full Details
My Workaround for the Missing Pieces
When the official manual wasn't cutting it, I ended up building my own set of notes by working through every odd problem myself and cross-referencing with older editions. The 8th edition solutions are available in the public domain through university archives and they overlap enough with the 9th that you can fill gaps. The problem numbering changed slightly between editions — Chapter 4 shifted by about three problems — but the core concepts didn't change. I also found that checking work against Wolfram Alpha's integral calculator for the specific contour integrals was surprisingly reliable. It won't show you the complex analysis steps, but it'll confirm whether your final numerical answer matches. Took about ten minutes to verify a twenty-minute manual integration in problem 6.2.9. One warning though — Wolfram Alpha defaults to principal values on some branch cut integrals, so if your problem specifies a non-principal branch, the output will look wrong even when your work is correct.
What to Watch Out For When Using Any Solution Set
The biggest issue isn't accuracy, it's learning. Students who use these manuals to check their work after attempting a problem get real value from it. Students who read through the solutions before doing the problems themselves usually end up thinking they understand the material when they don't. I saw this happen repeatedly. The problems in this textbook are designed to make you work through the mechanics of complex integration, and that mechanical practice is literally the point. Reading someone else's contour parametrization doesn't train your fingers to set one up correctly under exam conditions. Also, be aware that some third-party solution documents circulating online have typos in the final answers. I'd recommend always deriving your own answer first and then comparing. If your answer differs from a sketchy PDF you found on a random site, trust your derivation unless you can find a second independent source confirming the discrepancy.
Where to Actually Find Reliable Solutions
The most legitimate route is still the publisher's companion site or your institution's library. McGraw-Hill connects the textbook to a portal called Connect, which sometimes includes supplementary materials. Not all editions include the full solutions manual in that system, but the 9th edition does for the odd problems. If you're a student, talk to your professor about getting access. They usually have a copy on reserve or can add it to your course shell. For the even problems specifically, the textbook itself has some answers in the back, though they're just final results with no working shown. That's better than nothing for quick verification. My experience is that about sixty percent of the even-numbered answers in the back match the official manual, and the discrepancies tend to appear in the more computational problems where different rounding or intermediate step choices can lead to slightly different forms of the same answer.
